Identity formation

From Wikipedia, the free encyclopedia
https://en.wikipedia.org/wiki/Identity_formation

Identity formation, also called identity development or identity construction, is a complex process in which humans develop a clear and unique view of themselves and of their identity.

Self-concept, personality development, and values are all closely related to identity formation. Individuation is also a critical part of identity formation. Continuity and inner unity are healthy identity formation, while a disruption in either could be viewed and labeled as abnormal development; certain situations, like childhood trauma, can contribute to abnormal development. Specific factors also play a role in identity formation, such as race, ethnicity, and spirituality.

The concept of personal continuity, or personal identity, refers to an individual posing questions about themselves that challenge their original perception, like "Who am I?" The process defines individuals to others and themselves. Various factors make up a person's actual identity, including a sense of continuity, a sense of uniqueness from others, and a sense of affiliation based on their membership in various groups like family, ethnicity, and occupation. These group identities demonstrate the human need for affiliation or for people to define themselves in the eyes of others and themselves.

Identities are formed on many levels. The micro-level is self-definition, relations with people, and issues as seen from a personal or an individual perspective. The meso-level pertains to how identities are viewed, formed, and questioned by immediate communities and/or families. The macro-level are the connections among and individuals and issues from a national perspective. The global level connects individuals, issues, and groups at a worldwide level.

Identity is often described as finite and consisting of separate and distinct parts (e.g., family, cultural, personal, professional).

Theories

Many theories of development have aspects of identity formation included in them. Two theories directly address the process of identity formation: Erik Erikson's stages of psychosocial development (specifically the Identity versus Role Confusion stage), James Marcia's identity status theory, and Jeffrey Arnett's theories of identity formation in emerging adulthood.

Erikson's theory of identity vs. role confusion

Erikson's theory is that people experience different crises or conflicts throughout their lives in eight stages. Each stage occurs at a certain point in life and must be successfully resolved to progress to the next stage. The particular stage relevant to identity formation takes place during adolescence: Identity versus Role Confusion.

The Identity versus Role Confusion stage involves adolescents trying to figure out who they are in order to form a basic identity that they will build on throughout their life, especially concerning social and occupational identities. They ask themselves the existential questions: "Who am I?" and "What can I be?" They face the complexities of determining one's own identity. Erikson stated that this crisis is resolved with identity achievement, the point at which an individual has extensively considered various goals and values, accepting some and rejecting others, and understands who they are as a unique person. When an adolescent attains identity achievement, they are ready to enter the next stage of Erikson's theory, Intimacy versus Isolation, where they will form strong friendships and a sense of companionship with others.

If the Identity versus Role Confusion crisis is not positively resolved, an adolescent will face confusion about future plans, particularly their roles in adulthood. Failure to form one's own identity leads to failure to form a shared identity with others, which can lead to instability in many areas as an adult. The identity formation stage of Erik Erikson's theory of psychosocial development is a crucial stage in life.

Marcia's identity status theory

Marcia created a structural interview designed to classify adolescents into one of four statuses of identity. The statuses are used to describe and pinpoint the progression of an adolescent's identity formation process. In Marcia's theory, identity is operationally defined as whether an individual has explored various alternatives and made firm commitments to an occupation, religion, sexual orientation, and a set of political values.

The four identity statuses in James Marcia's theory are:

1. Identity Diffusion (also known as Role Confusion): The opposite of identity achievement. The individual has not resolved their identity crisis yet by failing to commit to any goals or values and establish a future life direction. In adolescents, this stage is characterized by disorganized thinking, procrastination, and avoidance of issues and actions.
2. Identity Foreclosure: This occurs when teenagers conform to an identity without exploring what suits them best. For instance, teenagers might follow the values and roles of their parents or cultural norms. They might also foreclose on a negative identity, or the direct opposite of their parents' values or cultural norms.
3. Identity Moratorium: This postpones identity achievement by providing temporary shelter. This status provides opportunities for exploration, either in breadth or in-depth. Examples of moratoria common in American society include college or the military.
4. Identity Achievement: This status is attained when the person has solved the identity issues by making commitments to goals, beliefs, and values after an extensive exploration of different areas.

Jeffrey Arnett's Theories on Identity Formation in Emerging Adulthood

Jeffrey Arnett's theory states that identity formation is most prominent in emerging adulthood, consisting of ages 18–25. Arnett holds that identity formation consists of indulging in different life opportunities and possibilities to eventually make important life decisions. He believes this phase of life includes a broad range of opportunities for identity formation, specifically in three different realms.

These three realms of identity exploration are:

1. Love: In emerging adulthood, individuals explore love to find a profound sense of intimacy. While trying to find love, individuals often explore their identity by focusing on questions such as: "Given the kind of person I am, what kind of person do I wish to have as a partner through life?"
2. Work: Work opportunities that people get involved in are now centered around the idea that they are preparing for careers that they might have throughout adulthood. Individuals explore their identity by asking themselves questions such as: "What kind of work am I good at?", "What kind of work would I find satisfying for the long term", or "What are my chance of getting a job in the field that seems to suit me best?"
3. Worldviews: It is common for those in the stage of emerging adulthood to attend college. There they may be exposed to different worldviews, compared to those they were raised in, and become open to altering their previous worldviews. Individuals who don't attend college also believe that as adult they should also decide what their beliefs and values are.

Self-concept

Self-concept, or self-identity, is the set of beliefs and ideas an individual has about themselves. Self-concept is different from self-consciousness, which is an awareness of one's self. Components of the self-concept include physical, psychological, and social attributes, which can be influenced by the individual's attitudes, habits, beliefs, and ideas; they cannot be condensed into the general concepts of self-image or self-esteem. Multiple types of identity come together within an individual and can be broken down into the following: cultural identity, professional identity, ethnic and national identity, religious identity, gender identity, and disability identity.

Cultural identity

Main article: Cultural identity

Cultural identity is formation of ideas an individual takes based on the culture they belong to. Cultural identity relates to but is not synonymous with identity politics. There are modern questions of culture that are transferred into questions of identity. Historical culture also influences individual identity, and as with modern cultural identity, individuals may pick and choose aspects of cultural identity, while rejecting or disowning other associated ideas.

Professional identity

Professional identity is the identification with a profession, exhibited by an aligning of roles, responsibilities, values, and ethical standards as accepted by the profession.

In business, professional identity is the professional self-concept that is founded upon attributes, values, and experiences. A professional identity is developed when there is a philosophy that is manifested in a distinct corporate culture – the corporate personality. A business professional is a person in a profession with certain types of skills that sometimes require formal training or education.

Career development encompasses the total dimensions of psychological, sociological, educational, physical, economic, and chance that alter a person's career practice across the lifespan. Career development also refers to the practices from a company or organization that enhance someone's career or encourages them to make practical career choices.

Training is a form of identity setting, since it not only affects knowledge but also affects a team member's self-concept. On the other hand, knowledge of the position introduces a new path of less effort to the trainee, which prolongs the effects of training and promotes a stronger self-concept. Other forms of identity setting in an organization include Business Cards, Specific Benefits by Role, and Task Forwarding.

Ethnic and national identity

An ethnic identity is an identification with a certain ethnicity, usually on the basis of a presumed common genealogy or ancestry. Recognition by others as a distinct ethnic group is often a contributing factor to developing this identity. Ethnic groups are also often united by common cultural, behavioral, linguistic, ritualistic, or religious traits.

Processes that result in the emergence of such identification are summarized as ethnogenesis. Various cultural studies and social theory investigate the question of cultural and ethnic identities. Cultural identity adheres to location, gender, race, history, nationality, sexual orientation, religious beliefs, and ethnicity.

National identity is an ethical and philosophical concept where all humans are divided into groups called nations. Members of a "nation" share a common identity and usually a common origin, in the sense of ancestry, parentage, or descent.

Religious identity

A religious identity is the set of beliefs and practices generally held by an individual, involving adherence to codified beliefs and rituals and study of ancestral or cultural traditions, writings, history, mythology, and faith and mystical experience. Religious identity refers to the personal practices related to communal faith along with rituals and communication stemming from such conviction. This identity formation begins with an association in the parents' religious contacts, and individuation requires that the person chooses the same or different religious identity than that of their parents.

Gender identity

In sociology, gender identity describes the gender with which a person identifies (i.e., whether one perceives oneself to be a man, a woman, outside of the gender binary), but can also be used to refer to the gender that other people attribute to the individual on the basis of what they know from gender role indications (social behavior, clothing, hairstyle, etc.). Gender identity may be affected by a variety of social structures, including the person's ethnic group, employment status, religion or irreligion, and family. It can also be biological in the sense of puberty.

Disability identity

Disability identity refers to the particular disabilities that an individual identifies with. This may be something as obvious as a paraplegic person identifying as such, or something less prominent such as a deaf person regarding themselves as part of a local, national, or global community of Deaf People Culture.

Disability identity is almost always determined by the particular disabilities that an individual is born with, though it may change later in life if an individual later becomes disabled or when an individual later discovers a previously overlooked disability (particularly applicable to mental disorders). In some rare cases, it may be influenced by exposure to disabled people as with body integrity dysphoria.

Political identity

Political identities often form the basis of public claims and mobilization of material and other resources for collective action. One theory that explores how this occurs is social movement theory. According to Charles Tilly, the interpretation of our relationship to others ("stories") create the rationale and construct of political identity. The capacity for action is constrained by material resources and sometimes perceptions that can be manipulated by using communication strategies that support the creation of illusory ties.

Interpersonal identity development

Interpersonal identity development comes from Marcia's Identity Status Theory, and refers to friendship, dating, gender roles, and recreation as tools to maturity in a psychosocial aspect of an individual.

Social relation can refer to a multitude of social interactions regulated by social norms between two or more people, with each having a social position and performing a social role. In a sociological hierarchy, social relation is more advanced than behavior, action, social behavior, social action, social contact, and social interaction. It forms the basis of concepts like social organization, social structure, social movement, and social system.

Interpersonal identity development is composed of three elements:

• Categorization: Assigning everyone into categories.
• Identification: Associating others with certain groups.
• Comparison: Comparing groups.

Interpersonal identity development allows an individual to question and examine various personality elements, such as ideas, beliefs, and behaviors. The actions or thoughts of others create social influences that change an individual. Examples of social influence can be seen in socialization and peer pressure, which can affect a person's behavior, thinking about one's self, and subsequent acceptance or rejection of how other people attempt to influence the individual. Interpersonal identity development occurs during exploratory self-analysis and self-evaluation, and ends at various times to establish an easy-to-understand and consolidative sense of self or identity.

Interaction

Main article: Hegelian dialectic

During interpersonal identity development, an exchange of propositions and counter-propositions occurs, resulting in a qualitative transformation of the individual. The aim of interpersonal identity development is to resolve the undifferentiated facets of an individual, which are found to be indistinguishable from others. Given this, and with other admissions, the individual is led to a contradiction between the self and others, and forces the withdrawal of the undifferentiated self as truth. To resolve the incongruence, the person integrates or rejects the encountered elements, which results in a new identity. During each of these exchanges, the individual must resolve the exchange before facing future ones. The exchanges are endless since the changing world constantly presents exchanges between individuals and thus allows individuals to redefine themselves constantly.

Collective identity

Collective identity is a sense of belonging to a group (the collective). If it is strong, an individual who identifies with the group will dedicate their lives to the group over individual identity: they will defend the views of the group and take risks for the group, often with little to no incentive or coercion. Collective identity often forms through a shared sense of interest, affiliation, or adversity. The cohesiveness of the collective identity goes beyond the community, as the collective experiences grief from the loss of a member.

Social support

Individuals gain a social identity and group identity from their affiliations in various groups, which include: family, ethnicity, education and occupational status, friendship, dating, and religion.

Family

One of the most important affiliations is that of the family, whether they be biological, extended, or even adoptive families. Each has its own influence on identity through the interaction that takes place between the family members and with the individual. Researchers and theorists state that an individual's identity (more specifically an adolescent's identity) is influenced by the people around them and the environment in which they live. If a family does not have integration, it is likely to cause identity diffusion (one of James Marcia's four identity statuses, where an individual has not made commitments and does not try to make them), and applies to both males and females.

Peer relationships

Morgan and Korobov performed a study in order to analyze the influence of same-sex friendships in the development of one's identity. This study involved the use of 24 same-sex college student friendship triads, consisting of 12 males and 12 females, with a total of 72 participants. Each triad was required to have known each other for a minimum of six months. A qualitative method was chosen, as it is the most appropriate in assessing the development of identity. Semi-structured group interviews took place, where the students were asked to reflect on stories and experiences concerning relationship problems. The results showed five common responses when assessing these relationship problems: joking about the relationship's problems, providing support, offering advice, relating others' experiences to their own similar experiences, and providing encouragement. The results concluded that adolescents actively construct their identities through common themes of conversation between same-sex friendships; in this case, involving relationship issues. The common themes of conversation that close peers seem to engage in helping to further their identity formation in life.

Influences on identity

Cognitive influences

Cognitive development influences identity formation. When adolescents are able to think abstractly and reason logically, they have an easier time exploring and contemplating possible identities. When an adolescent has advanced cognitive development and maturity, they tend to resolve identity issues more so than age mates that are less cognitively developed. When identity issues are solved quicker and better, there is more time and effort put into developing that identity.

Scholastic influences

Adolescents that have a post-secondary education tend to make more concrete goals and stable occupational commitments. Going to college or university can influence identity formation in a productive way. The opposite can also be true, where identity influences education and academics. Education's effect on identity can be beneficial for the individual's identity; the individual becomes educated on different approaches and paths to take in the process of identity formation.

Sociocultural influences

Sociocultural influences are those of a broader social and historical context. For example, in the past, adolescents would likely just adopt the job or religious beliefs that were expected of them or that were akin to their parents. Today, adolescents have more resources to explore identity choices and more options for commitments. This influence is becoming less significant due to the growing acceptance of identity options that were once less accepted. Many of the identity options from the past are becoming unrecognized and less popular today. The changing sociocultural situation is forcing individuals to develop a unique identity based on their own aspirations. Sociocultural influences play a different role in identity formation now than they have in the past.

Parenting influences

The type of relationship that adolescents have with their parents has a significant role in identity formation. For example, when there is a solid and positive relationship between parents and adolescents, they are more likely to feel freedom in exploring identity options for themselves. A study found that for boys and girls, identity formation is positively influenced by parental involvement, specifically in the areas of support, social monitoring, and school monitoring. In contrast, when the relationship is not as close and the fear of rejection or discontentment from the parent or other guardians is present, they are more likely to feel less confident in forming a separate identity from their parents.

Cyber-socializing and the Internet

See also: Online identity, Cyberpsychology, and Sociology of the Internet

The Internet is becoming an extension of the expressive dimension of adolescence. On the Internet, youth talk about their lives and concerns, design the content that they make available to others, and assess the reactions of others to it in the form of optimized and electronically mediated social approval. When connected, youth speak of their daily routines and lives. With each post, image or video they upload, they can ask themselves who they are and try out profiles that differ from the ones they practice in the "real" world.

orthogonality

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Orthogonality

The line segments AB and CD are orthogonal to each other.

In mathematics, orthogonality is the generalization of the geometric notion of perpendicularity.

Orthogonality is also used with various meanings that are often weakly related or not related at all with the mathematical meanings.

Etymology

The word comes from the Ancient Greek ὀρθός (orthós), meaning "upright", and γωνία (gōnía), meaning "angle".

The Ancient Greek ὀρθογώνιον (orthogṓnion) and Classical Latin orthogonium originally denoted a rectangle. Later, they came to mean a right triangle. In the 12th century, the post-classical Latin word orthogonalis came to mean a right angle or something related to a right angle.

Mathematics

In mathematics, orthogonality is the generalization of the geometric notion of perpendicularity to the linear algebra of bilinear forms.

Two elements u and v of a vector space with bilinear form ${\displaystyle B}$ are orthogonal when ${\displaystyle B(\mathbf{u},\mathbf{v})=0}$. Depending on the bilinear form, the vector space may contain non-zero self-orthogonal vectors. In the case of function spaces, families of orthogonal functions are used to form an orthogonal basis.

The concept has been used in the context of orthogonal functions, orthogonal polynomials, and combinatorics.

Orthogonality and rotation of coordinate systems compared between left: Euclidean space through circular angle ϕ, right: in Minkowski spacetime through hyperbolic angle ϕ (red lines labelled c denote the worldlines of a light signal, a vector is orthogonal to itself if it lies on this line).

Physics

Optics

In optics, polarization states are said to be orthogonal when they propagate independently of each other, as in vertical and horizontal linear polarization or right- and left-handed circular polarization.

Special relativity

In special relativity, a time axis determined by a rapidity of motion is hyperbolic-orthogonal to a space axis of simultaneous events, also determined by the rapidity. The theory features relativity of simultaneity.

Hyperbolic orthogonality

Euclidean orthogonality is preserved by rotation in the left diagram; hyperbolic orthogonality with respect to hyperbola (B) is preserved by hyperbolic rotation in the right diagram.

In geometry, the relation of hyperbolic orthogonality between two lines separated by the asymptotes of a hyperbola is a concept used in special relativity to define simultaneous events. Two events will be simultaneous when they are on a line hyperbolically orthogonal to a particular time line. This dependence on a certain time line is determined by velocity, and is the basis for the relativity of simultaneity.

Quantum mechanics

Main article: Quantum mechanics

In quantum mechanics, a sufficient (but not necessary) condition that two eigenstates of a Hermitian operator, ${\displaystyle {\psi }_m}$ and ${\displaystyle {\psi }_n}$, are orthogonal is that they correspond to different eigenvalues. This means, in Dirac notation, that ${\displaystyle ⟨{\psi }_m|{\psi }_n⟩=0}$ if ${\displaystyle {\psi }_m}$ and ${\displaystyle {\psi }_n}$ correspond to different eigenvalues. This follows from the fact that Schrödinger's equation is a Sturm–Liouville equation (in Schrödinger's formulation) or that observables are given by Hermitian operators (in Heisenberg's formulation).

Art

In art, the perspective (imaginary) lines pointing to the vanishing point are referred to as "orthogonal lines". The term "orthogonal line" often has a quite different meaning in the literature of modern art criticism. Many works by painters such as Piet Mondrian and Burgoyne Diller are noted for their exclusive use of "orthogonal lines" — not, however, with reference to perspective, but rather referring to lines that are straight and exclusively horizontal or vertical, forming right angles where they intersect. For example, an essay at the web site of the Thyssen-Bornemisza Museum states that "Mondrian ... dedicated his entire oeuvre to the investigation of the balance between orthogonal lines and primary colours." Archived 2009-01-31 at the Wayback Machine

Computer science

Further information: Orthogonality (programming) and Orthogonal instruction set

Orthogonality in programming language design is the ability to use various language features in arbitrary combinations with consistent results. This usage was introduced by Van Wijngaarden in the design of Algol 68:

The number of independent primitive concepts has been minimized in order that the language be easy to describe, to learn, and to implement. On the other hand, these concepts have been applied “orthogonally” in order to maximize the expressive power of the language while trying to avoid deleterious superfluities.

Orthogonality is a system design property which guarantees that modifying the technical effect produced by a component of a system neither creates nor propagates side effects to other components of the system. Typically this is achieved through the separation of concerns and encapsulation, and it is essential for feasible and compact designs of complex systems. The emergent behavior of a system consisting of components should be controlled strictly by formal definitions of its logic and not by side effects resulting from poor integration, i.e., non-orthogonal design of modules and interfaces. Orthogonality reduces testing and development time because it is easier to verify designs that neither cause side effects nor depend on them.

Orthogonal instruction set

Main article: Orthogonal instruction set

An instruction set is said to be orthogonal if it lacks redundancy (i.e., there is only a single instruction that can be used to accomplish a given task) and is designed such that instructions can use any register in any addressing mode. This terminology results from considering an instruction as a vector whose components are the instruction fields. One field identifies the registers to be operated upon and another specifies the addressing mode. An orthogonal instruction set uniquely encodes all combinations of registers and addressing modes.

Telecommunications

In telecommunications, multiple access schemes are orthogonal when an ideal receiver can completely reject arbitrarily strong unwanted signals from the desired signal using different basis functions. One such scheme is time-division multiple access (TDMA), where the orthogonal basis functions are nonoverlapping rectangular pulses ("time slots").

Orthogonal frequency-division multiplexing

Main article: Orthogonal frequency-division multiplexing

Another scheme is orthogonal frequency-division multiplexing (OFDM), which refers to the use, by a single transmitter, of a set of frequency multiplexed signals with the exact minimum frequency spacing needed to make them orthogonal so that they do not interfere with each other. Well known examples include (a, g, and n) versions of 802.11 Wi-Fi; WiMAX; ITU-T G.hn, DVB-T, the terrestrial digital TV broadcast system used in most of the world outside North America; and DMT (Discrete Multi Tone), the standard form of ADSL.

In OFDM, the subcarrier frequencies are chosen so that the subcarriers are orthogonal to each other, meaning that crosstalk between the subchannels is eliminated and intercarrier guard bands are not required. This greatly simplifies the design of both the transmitter and the receiver. In conventional FDM, a separate filter for each subchannel is required.

Statistics, econometrics, and economics

When performing statistical analysis, independent variables that affect a particular dependent variable are said to be orthogonal if they are uncorrelated, since the covariance forms an inner product. In this case the same results are obtained for the effect of any of the independent variables upon the dependent variable, regardless of whether one models the effects of the variables individually with simple regression or simultaneously with multiple regression. If correlation is present, the factors are not orthogonal and different results are obtained by the two methods. This usage arises from the fact that if centered by subtracting the expected value (the mean), uncorrelated variables are orthogonal in the geometric sense discussed above, both as observed data (i.e., vectors) and as random variables (i.e., density functions). One econometric formalism that is alternative to the maximum likelihood framework, the Generalized Method of Moments, relies on orthogonality conditions. In particular, the Ordinary Least Squares estimator may be easily derived from an orthogonality condition between the explanatory variables and model residuals.

Taxonomy

In taxonomy, an orthogonal classification is one in which no item is a member of more than one group, that is, the classifications are mutually exclusive.

Chemistry and biochemistry

In chemistry and biochemistry, an orthogonal interaction occurs when there are two pairs of substances and each substance can interact with their respective partner, but does not interact with either substance of the other pair. For example, DNA has two orthogonal pairs: cytosine and guanine form a base-pair, and adenine and thymine form another base-pair, but other base-pair combinations are strongly disfavored. As a chemical example, tetrazine reacts with transcyclooctene and azide reacts with cyclooctyne without any cross-reaction, so these are mutually orthogonal reactions, and so, can be performed simultaneously and selectively.

Organic synthesis

Main articles: Organic synthesis and Orthogonal protection

In organic synthesis, orthogonal protection is a strategy allowing the deprotection of functional groups independently of each other.

Bioorthogonal chemistry

The term bioorthogonal chemistry refers to any chemical reaction that can occur inside of living systems without interfering with native biochemical processes. The term was coined by Carolyn R. Bertozzi in 2003. Since its introduction, the concept of the bioorthogonal reaction has enabled the study of biomolecules such as glycans, proteins, and lipids in real time in living systems without cellular toxicity. A number of chemical ligation strategies have been developed that fulfill the requirements of bioorthogonality, including the 1,3-dipolar cycloaddition between azides and cyclooctynes (also termed copper-free click chemistry), between nitrones and cyclooctynes, oxime/hydrazone formation from aldehydes and ketones, the tetrazine ligation, the isocyanide-based click reaction, and most recently, the quadricyclane ligation.

Supramolecular chemistry

Main article: Supramolecular chemistry

In supramolecular chemistry the notion of orthogonality refers to the possibility of two or more supramolecular, often non-covalent, interactions being compatible; reversibly forming without interference from the other.

Analytical chemistry

Main article: Analytical chemistry

In analytical chemistry, analyses are "orthogonal" if they make a measurement or identification in completely different ways, thus increasing the reliability of the measurement. Orthogonal testing thus can be viewed as "cross-checking" of results, and the "cross" notion corresponds to the etymologic origin of orthogonality. Orthogonal testing is often required as a part of a new drug application.

System reliability

In the field of system reliability orthogonal redundancy is that form of redundancy where the form of backup device or method is completely different from the prone to error device or method. The failure mode of an orthogonally redundant back-up device or method does not intersect with and is completely different from the failure mode of the device or method in need of redundancy to safeguard the total system against catastrophic failure.

Neuroscience

In neuroscience, a sensory map in the brain which has overlapping stimulus coding (e.g. location and quality) is called an orthogonal map.

Philosophy

In philosophy, two topics, authors, or pieces of writing are said to be "orthogonal" to each other when they do not substantively cover what could be considered potentially overlapping or competing claims. Thus, texts in philosophy can either support and complement one another, they can offer competing explanations or systems, or they can be orthogonal to each other in cases where the scope, content, and purpose of the pieces of writing are entirely unrelated.

Gaming

See also: von Neumann neighborhood

In board games such as chess which feature a grid of squares, 'orthogonal' is used to mean "in the same row/'rank' or column/'file'". This is the counterpart to squares which are "diagonally adjacent". In the ancient Chinese board game Go a player can capture the stones of an opponent by occupying all orthogonally adjacent points.

Other examples

Stereo vinyl records encode both the left and right stereo channels in a single groove. The V-shaped groove in the vinyl has walls that are 90 degrees to each other, with variations in each wall separately encoding one of the two analogue channels that make up the stereo signal. The cartridge senses the motion of the stylus following the groove in two orthogonal directions: 45 degrees from vertical to either side. A pure horizontal motion corresponds to a mono signal, equivalent to a stereo signal in which both channels carry identical (in-phase) signals.

Inner product space

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Inner_product_space

Geometric interpretation of the angle between two vectors defined using an inner product

Scalar product spaces, over any field, have "scalar products" that are symmetrical and linear in the first argument. Hermitian product spaces are restricted to the field of complex numbers and have "Hermitian products" that are conjugate-symmetrical and linear in the first argument. Inner product spaces may be defined over any field, having "inner products" that are linear in the first argument, conjugate-symmetrical, and positive-definite. Unlike inner products, scalar products and Hermitian products need not be positive-definite.

In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, often denoted with angle brackets such as in ${\displaystyle ⟨a,b⟩}$. Inner products allow formal definitions of intuitive geometric notions, such as lengths, angles, and orthogonality (zero inner product) of vectors. Inner product spaces generalize Euclidean vector spaces, in which the inner product is the dot product or scalar product of Cartesian coordinates. Inner product spaces of infinite dimension are widely used in functional analysis. Inner product spaces over the field of complex numbers are sometimes referred to as unitary spaces. The first usage of the concept of a vector space with an inner product is due to Giuseppe Peano, in 1898.

An inner product naturally induces an associated norm, (denoted ${\displaystyle |x|}$ and ${\displaystyle |y|}$ in the picture); so, every inner product space is a normed vector space. If this normed space is also complete (that is, a Banach space) then the inner product space is a Hilbert space. If an inner product space H is not a Hilbert space, it can be extended by completion to a Hilbert space ${\displaystyle \overline{H}.}$ This means that ${\displaystyle H}$ is a linear subspace of ${\displaystyle \overline{H},}$ the inner product of ${\displaystyle H}$ is the restriction of that of ${\displaystyle \overline{H},}$ and ${\displaystyle H}$ is dense in ${\displaystyle \overline{H}}$ for the topology defined by the norm.

Definition

In this article, F denotes a field that is either the real numbers ${\displaystyle \mathbb{R},}$ or the complex numbers ${\displaystyle \mathbb{C}.}$ A scalar is thus an element of F. A bar over an expression representing a scalar denotes the complex conjugate of this scalar. A zero vector is denoted ${\displaystyle \mathbf{0}}$ for distinguishing it from the scalar 0.

An inner product space is a vector space V over the field F together with an inner product, that is, a map

${\displaystyle ⟨\cdot ,\cdot ⟩:V\times V\to F}$

that satisfies the following three properties for all vectors ${\displaystyle x,y,z\in V}$ and all scalars ${\displaystyle a,b\in F}$.

• Conjugate symmetry: $${\displaystyle ⟨x,y⟩=\overline{⟨y,x⟩}.}$$ As $a=\overline{a}$ if and only if ${\displaystyle a}$ is real, conjugate symmetry implies that ${\displaystyle ⟨x,x⟩}$ is always a real number. If F is ${\displaystyle \mathbb{R}}$, conjugate symmetry is just symmetry.
• Linearity in the first argument:^{[Note 1]} $${\displaystyle ⟨ax+by,z⟩=a⟨x,z⟩+b⟨y,z⟩.}$$
• Positive-definiteness: if ${\displaystyle x}$ is not zero, then $${\displaystyle ⟨x,x⟩>0}$$ (conjugate symmetry implies that ${\displaystyle ⟨x,x⟩}$ is real).

If the positive-definiteness condition is replaced by merely requiring that ${\displaystyle ⟨x,x⟩\ge 0}$ for all ${\displaystyle x}$, then one obtains the definition of positive semi-definite Hermitian form. A positive semi-definite Hermitian form ${\displaystyle ⟨\cdot ,\cdot ⟩}$ is an inner product if and only if for all ${\displaystyle x}$, if ${\displaystyle ⟨x,x⟩=0}$ then ${\displaystyle x=\mathbf{0}}$.

Basic properties

In the following properties, which result almost immediately from the definition of an inner product, x, y and z are arbitrary vectors, and a and b are arbitrary scalars.

• ${\displaystyle ⟨\mathbf{0},x⟩=⟨x,\mathbf{0}⟩=0.}$
• ${\displaystyle ⟨x,x⟩}$ is real and nonnegative.
• ${\displaystyle ⟨x,x⟩=0}$ if and only if ${\displaystyle x=\mathbf{0}.}$
• ${\displaystyle ⟨x,ay+bz⟩=\overline{a}⟨x,y⟩+\overline{b}⟨x,z⟩.}$
  This implies that an inner product is a sesquilinear form.
• ${\displaystyle ⟨x+y,x+y⟩=⟨x,x⟩+2\mathrm{Re}(⟨x,y⟩)+⟨y,y⟩,}$ where ${\displaystyle \mathrm{Re}}$
  denotes the real part of its argument.

Over ${\displaystyle \mathbb{R}}$, conjugate-symmetry reduces to symmetry, and sesquilinearity reduces to bilinearity. Hence an inner product on a real vector space is a positive-definite symmetric bilinear form. The binomial expansion of a square becomes

${\displaystyle ⟨x+y,x+y⟩=⟨x,x⟩+2⟨x,y⟩+⟨y,y⟩.}$

Convention variant

Some authors, especially in physics and matrix algebra, prefer to define inner products and sesquilinear forms with linearity in the second argument rather than the first. Then the first argument becomes conjugate linear, rather than the second. Bra-ket notation in quantum mechanics also uses slightly different notation, i.e. ${\displaystyle ⟨\cdot |\cdot ⟩}$, where ${\displaystyle ⟨x|y⟩:=\left(y,x\right)}$.

Notation

Several notations are used for inner products, including ${\displaystyle ⟨\cdot ,\cdot ⟩}$, ${\displaystyle \left(\cdot ,\cdot \right)}$, ${\displaystyle ⟨\cdot |\cdot ⟩}$ and ${\displaystyle \left(\cdot |\cdot \right)}$, as well as the usual dot product.

Some examples

Real and complex numbers

Among the simplest examples of inner product spaces are ${\displaystyle \mathbb{R}}$ and ${\displaystyle \mathbb{C}.}$ The real numbers ${\displaystyle \mathbb{R}}$ are a vector space over ${\displaystyle \mathbb{R}}$ that becomes an inner product space with arithmetic multiplication as its inner product: $${\displaystyle ⟨x,y⟩:=xy\text{ for }x,y\in \mathbb{R}.}$$

The complex numbers ${\displaystyle \mathbb{C}}$ are a vector space over ${\displaystyle \mathbb{C}}$ that becomes an inner product space with the inner product $${\displaystyle ⟨x,y⟩:=x\overline{y}\text{ for }x,y\in \mathbb{C}.}$$ Unlike with the real numbers, the assignment ${\displaystyle (x,y)\mapsto xy}$ does not define a complex inner product on ${\displaystyle \mathbb{C}.}$

Euclidean vector space

More generally, the real ${\displaystyle n}$-space ${\displaystyle {\mathbb{R}}^n}$ with the dot product is an inner product space, an example of a Euclidean vector space. $${\displaystyle ⟨\left[\begin{array}{c}{x}_1\\ ⋮\\ x_n\end{array}\right],\left[\begin{array}{c}{y}_1\\ ⋮\\ y_n\end{array}\right]⟩=x^{\mathsf{\text{T}}}y=\sum _{i=1}^n{x}_i{y}_i=x_1{y}_1+\cdots +x_n{y}_n,}$$ where ${\displaystyle x^{\mathrm{T}}}$ is the transpose of ${\displaystyle x.}$

A function ${\displaystyle ⟨\cdot ,\cdot ⟩:{\mathbb{R}}^n\times {\mathbb{R}}^n\to \mathbb{R}}$ is an inner product on ${\displaystyle {\mathbb{R}}^n}$ if and only if there exists a symmetric positive-definite matrix ${\displaystyle \mathbf{M}}$ such that ${\displaystyle ⟨x,y⟩=x^{\mathrm{T}}\mathbf{M}y}$ for all ${\displaystyle x,y\in {\mathbb{R}}^n.}$ If ${\displaystyle \mathbf{M}}$ is the identity matrix then ${\displaystyle ⟨x,y⟩=x^{\mathrm{T}}\mathbf{M}y}$ is the dot product. For another example, if ${\displaystyle n=2}$ and ${\displaystyle \mathbf{M}=\left[\begin{array}{cc}a& b\\ b& d\end{array}\right]}$ is positive-definite (which happens if and only if ${\displaystyle det\mathbf{M}=ad-b^2>0}$ and one/both diagonal elements are positive) then for any ${\displaystyle x:={\left[x_1,x_2\right]}^{\mathrm{T}},y:={\left[y_1,y_2\right]}^{\mathrm{T}}\in {\mathbb{R}}^2,}$ $${\displaystyle ⟨x,y⟩:=x^{\mathrm{T}}\mathbf{M}y=\left[x_1,x_2\right]\left[\begin{array}{cc}a& b\\ b& d\end{array}\right]\left[\begin{array}{c}{y}_1\\ y_2\end{array}\right]=a{x}_1{y}_1+b{x}_1{y}_2+b{x}_2{y}_1+d{x}_2{y}_2.}$$ As mentioned earlier, every inner product on ${\displaystyle {\mathbb{R}}^2}$ is of this form (where ${\displaystyle b\in \mathbb{R},a>0}$ and ${\displaystyle d>0}$ satisfy ${\displaystyle ad>b^2}$).

Complex coordinate space

The general form of an inner product on ${\displaystyle {\mathbb{C}}^n}$ is known as the Hermitian form and is given by $${\displaystyle ⟨x,y⟩=y^{†}\mathbf{M}x=\overline{x^{†}\mathbf{M}y},}$$ where ${\displaystyle M}$ is any Hermitian positive-definite matrix and ${\displaystyle y^{†}}$ is the conjugate transpose of ${\displaystyle y.}$ For the real case, this corresponds to the dot product of the results of directionally-different scaling of the two vectors, with positive scale factors and orthogonal directions of scaling. It is a weighted-sum version of the dot product with positive weights—up to an orthogonal transformation.

Hilbert space

The article on Hilbert spaces has several examples of inner product spaces, wherein the metric induced by the inner product yields a complete metric space. An example of an inner product space which induces an incomplete metric is the space ${\displaystyle C([a,b])}$ of continuous complex valued functions ${\displaystyle f}$ and ${\displaystyle g}$ on the interval ${\displaystyle [a,b].}$ The inner product is $${\displaystyle ⟨f,g⟩={\int }_a^{b}f(t)\overline{g(t)}\mathrm{d}t.}$$ This space is not complete; consider for example, for the interval [−1, 1] the sequence of continuous "step" functions, ${\displaystyle \{f_k{\}}_k,}$ defined by: $${\displaystyle f_k(t)=\{\begin{array}{ll}0& t\in [-1,0]\\ 1& t\in \left[\frac{1}{k},1\right]\\ kt& t\in \left(0,\frac{1}{k}\right)\end{array}}$$

This sequence is a Cauchy sequence for the norm induced by the preceding inner product, which does not converge to a continuous function.

Random variables

For real random variables ${\displaystyle X}$ and ${\displaystyle Y,}$ the expected value of their product $${\displaystyle ⟨X,Y⟩=\mathbb{E}[XY]}$$ is an inner product. In this case, ${\displaystyle ⟨X,X⟩=0}$ if and only if ${\displaystyle \mathbb{P}[X=0]=1}$ (that is, ${\displaystyle X=0}$ almost surely), where ${\displaystyle \mathbb{P}}$ denotes the probability of the event. This definition of expectation as inner product can be extended to random vectors as well.

Complex matrices

The inner product for complex square matrices of the same size is the Frobenius inner product ${\displaystyle ⟨A,B⟩:=\mathrm{tr}\left(A{B}^{†}\right)}$. Since trace and transposition are linear and the conjugation is on the second matrix, it is a sesquilinear operator. We further get Hermitian symmetry by, $${\displaystyle ⟨A,B⟩=\mathrm{tr}\left(A{B}^{†}\right)=\overline{\mathrm{tr}\left(B{A}^{†}\right)}=\overline{⟨B,A⟩}}$$ Finally, since for ${\displaystyle A}$ nonzero, ${\displaystyle ⟨A,A⟩=\sum _{ij}{\left|A_{ij}\right|}^2>0}$, we get that the Frobenius inner product is positive definite too, and so is an inner product.

Vector spaces with forms

On an inner product space, or more generally a vector space with a nondegenerate form (hence an isomorphism ${\displaystyle V\to V^{\ast }}$), vectors can be sent to covectors (in coordinates, via transpose), so that one can take the inner product and outer product of two vectors—not simply of a vector and a covector.

Basic results, terminology, and definitions

Norm properties

Every inner product space induces a norm, called its canonical norm, that is defined by $${\displaystyle \Vert x\Vert =\sqrt{⟨x,x⟩}.}$$ With this norm, every inner product space becomes a normed vector space.

So, every general property of normed vector spaces applies to inner product spaces. In particular, one has the following properties:

Absolute homogeneity

$${\displaystyle \Vert ax\Vert =|a|\Vert x\Vert }$$ for every ${\displaystyle x\in V}$ and ${\displaystyle a\in F}$ (this results from ${\displaystyle ⟨ax,ax⟩=a\overline{a}⟨x,x⟩}$).

Triangle inequality

$${\displaystyle \Vert x+y\Vert \le \Vert x\Vert +\Vert y\Vert }$$ for ${\displaystyle x,y\in V.}$ These two properties show that one has indeed a norm.

Cauchy–Schwarz inequality

$${\displaystyle |⟨x,y⟩|\le \Vert x\Vert \Vert y\Vert }$$ for every ${\displaystyle x,y\in V,}$ with equality if and only if ${\displaystyle x}$ and ${\displaystyle y}$ are linearly dependent.

Parallelogram law

$${\displaystyle \Vert x+y{\Vert }^2+\Vert x-y{\Vert }^2=2\Vert x{\Vert }^2+2\Vert y{\Vert }^2}$$ for every ${\displaystyle x,y\in V.}$ The parallelogram law is a necessary and sufficient condition for a norm to be defined by an inner product.

Polarization identity

$${\displaystyle \Vert x+y{\Vert }^2=\Vert x{\Vert }^2+\Vert y{\Vert }^2+2\mathrm{Re}⟨x,y⟩}$$ for every ${\displaystyle x,y\in V.}$ The inner product can be retrieved from the norm by the polarization identity, since its imaginary part is the real part of ${\displaystyle ⟨x,iy⟩.}$

Ptolemy's inequality

$${\displaystyle \Vert x-y\Vert \Vert z\Vert +\Vert y-z\Vert \Vert x\Vert \ge \Vert x-z\Vert \Vert y\Vert }$$ for every ${\displaystyle x,y,z\in V.}$ Ptolemy's inequality is a necessary and sufficient condition for a seminorm to be the norm defined by an inner product.

Orthogonality

Orthogonality

Two vectors ${\displaystyle x}$ and ${\displaystyle y}$ are said to be orthogonal, often written ${\displaystyle x\perp y,}$ if their inner product is zero, that is, if ${\displaystyle ⟨x,y⟩=0.}$
This happens if and only if ${\displaystyle \Vert x\Vert \le \Vert x+sy\Vert }$ for all scalars ${\displaystyle s,}$ and if and only if the real-valued function ${\displaystyle f(s):=\Vert x+sy{\Vert }^2-\Vert x{\Vert }^2}$ is non-negative. (This is a consequence of the fact that, if ${\displaystyle y\ne 0}$ then the scalar ${\displaystyle s_0=-\frac{\overline{⟨x,y⟩}}{\Vert y{\Vert }^2}}$ minimizes ${\displaystyle f}$ with value ${\displaystyle f\left(s_0\right)=-\frac{|⟨x,y⟩{|}^2}{\Vert y{\Vert }^2},}$ which is always non positive).
For a complex inner product space ${\displaystyle H,}$ a linear operator ${\displaystyle T:V\to V}$ is identically ${\displaystyle 0}$ if and only if ${\displaystyle x\perp Tx}$ for every ${\displaystyle x\in V.}$ This is not true in general for real inner product spaces, as it is a consequence of conjugate symmetry being distinct from symmetry for complex inner products. A counterexample in a real inner product space is ${\displaystyle T}$ a 90° rotation in ${\displaystyle {\mathbb{R}}^2}$, which maps every vector to an orthogonal vector but is not identically ${\displaystyle 0}$.

Orthogonal complement

The orthogonal complement of a subset ${\displaystyle C\subseteq V}$ is the set ${\displaystyle C^{\mathrm{\perp }}}$ of the vectors that are orthogonal to all elements of C; that is, $${\displaystyle C^{\mathrm{\perp }}:=\{y\in V:⟨y,c⟩=0\text{ for all }c\in C\}.}$$ This set ${\displaystyle C^{\mathrm{\perp }}}$ is always a closed vector subspace of ${\displaystyle V}$ and if the closure ${\displaystyle {\mathrm{cl}}_{V}C}$ of ${\displaystyle C}$ in ${\displaystyle V}$ is a vector subspace then ${\displaystyle {\mathrm{cl}}_{V}C={\left(C^{\mathrm{\perp }}\right)}^{\mathrm{\perp }}.}$

Pythagorean theorem

If ${\displaystyle x}$ and ${\displaystyle y}$ are orthogonal, then $${\displaystyle \Vert x{\Vert }^2+\Vert y{\Vert }^2=\Vert x+y{\Vert }^2.}$$ This may be proved by expressing the squared norms in terms of the inner products, using additivity for expanding the right-hand side of the equation.
The name Pythagorean theorem arises from the geometric interpretation in Euclidean geometry.

Parseval's identity

An induction on the Pythagorean theorem yields: if ${\displaystyle x_1,\dots ,x_n}$ are pairwise orthogonal, then $${\displaystyle \sum _{i=1}^n\Vert x_i{\Vert }^2={\Vert \sum _{i=1}^n{x}_i\Vert }^2.}$$

Angle

When ${\displaystyle ⟨x,y⟩}$ is a real number then the Cauchy–Schwarz inequality implies that $\frac{⟨x,y⟩}{\Vert x\Vert \Vert y\Vert }\in [-1,1],$ and thus that $${\displaystyle \mathrm{\angle }(x,y)=\arccos \frac{⟨x,y⟩}{\Vert x\Vert \Vert y\Vert },}$$ is a real number. This allows defining the (non oriented) angle of two vectors in modern definitions of Euclidean geometry in terms of linear algebra. This is also used in data analysis, under the name "cosine similarity", for comparing two vectors of data.

Real and complex parts of inner products

Suppose that ${\displaystyle ⟨\cdot ,\cdot ⟩}$ is an inner product on ${\displaystyle V}$ (so it is antilinear in its second argument). The polarization identity shows that the real part of the inner product is $${\displaystyle \mathrm{Re}⟨x,y⟩=\frac{1}{4}\left(\Vert x+y{\Vert }^2-\Vert x-y{\Vert }^2\right).}$$

If ${\displaystyle V}$ is a real vector space then $${\displaystyle ⟨x,y⟩=\mathrm{Re}⟨x,y⟩=\frac{1}{4}\left(\Vert x+y{\Vert }^2-\Vert x-y{\Vert }^2\right)}$$ and the imaginary part (also called the complex part) of ${\displaystyle ⟨\cdot ,\cdot ⟩}$ is always ${\displaystyle 0.}$

Assume for the rest of this section that ${\displaystyle V}$ is a complex vector space. The polarization identity for complex vector spaces shows that

${\displaystyle \begin{array}{rl}⟨x,y⟩& =\frac{1}{4}\left(\Vert x+y{\Vert }^2-\Vert x-y{\Vert }^2+i\Vert x+iy{\Vert }^2-i\Vert x-iy{\Vert }^2\right)\\ & =\mathrm{Re}⟨x,y⟩+i\mathrm{Re}⟨x,iy⟩.\end{array}}$

The map defined by ${\displaystyle ⟨x\mid y⟩=⟨y,x⟩}$ for all ${\displaystyle x,y\in V}$ satisfies the axioms of the inner product except that it is antilinear in its first, rather than its second, argument. The real part of both ${\displaystyle ⟨x\mid y⟩}$ and ${\displaystyle ⟨x,y⟩}$ are equal to ${\displaystyle \mathrm{Re}⟨x,y⟩}$ but the inner products differ in their complex part:

${\displaystyle \begin{array}{rl}⟨x\mid y⟩& =\frac{1}{4}\left(\Vert x+y{\Vert }^2-\Vert x-y{\Vert }^2-i\Vert x+iy{\Vert }^2+i\Vert x-iy{\Vert }^2\right)\\ & =\mathrm{Re}⟨x,y⟩-i\mathrm{Re}⟨x,iy⟩.\end{array}}$

The last equality is similar to the formula expressing a linear functional in terms of its real part.

These formulas show that every complex inner product is completely determined by its real part. Moreover, this real part defines an inner product on ${\displaystyle V,}$ considered as a real vector space. There is thus a one-to-one correspondence between complex inner products on a complex vector space ${\displaystyle V,}$ and real inner products on ${\displaystyle V.}$

For example, suppose that ${\displaystyle V={\mathbb{C}}^n}$ for some integer ${\displaystyle n>0.}$ When ${\displaystyle V}$ is considered as a real vector space in the usual way (meaning that it is identified with the ${\displaystyle 2n-}$dimensional real vector space ${\displaystyle {\mathbb{R}}^{2n},}$ with each ${\displaystyle \left(a_1+i{b}_1,\dots ,a_n+i{b}_n\right)\in {\mathbb{C}}^n}$ identified with ${\displaystyle \left(a_1,b_1,\dots ,a_n,b_n\right)\in {\mathbb{R}}^{2n}}$), then the dot product ${\displaystyle x\cdot y=\left(x_1,\dots ,x_{2n}\right)\cdot \left(y_1,\dots ,y_{2n}\right):=x_1{y}_1+\cdots +x_{2n}{y}_{2n}}$ defines a real inner product on this space. The unique complex inner product ${\displaystyle ⟨\cdot ,\cdot ⟩}$ on ${\displaystyle V={\mathbb{C}}^n}$ induced by the dot product is the map that sends ${\displaystyle c=\left(c_1,\dots ,c_n\right),d=\left(d_1,\dots ,d_n\right)\in {\mathbb{C}}^n}$ to ${\displaystyle ⟨c,d⟩:=c_1\overline{d_1}+\cdots +c_n\overline{d_n}}$ (because the real part of this map ${\displaystyle ⟨\cdot ,\cdot ⟩}$ is equal to the dot product).

Real vs. complex inner products

Let ${\displaystyle V_{\mathbb{R}}}$ denote ${\displaystyle V}$ considered as a vector space over the real numbers rather than complex numbers. The real part of the complex inner product ${\displaystyle ⟨x,y⟩}$ is the map ${\displaystyle ⟨x,y{⟩}_{\mathbb{R}}=\mathrm{Re}⟨x,y⟩:V_{\mathbb{R}}\times V_{\mathbb{R}}\to \mathbb{R},}$ which necessarily forms a real inner product on the real vector space ${\displaystyle V_{\mathbb{R}}.}$ Every inner product on a real vector space is a bilinear and symmetric map.

For example, if ${\displaystyle V=\mathbb{C}}$ with inner product ${\displaystyle ⟨x,y⟩=x\overline{y},}$ where ${\displaystyle V}$ is a vector space over the field ${\displaystyle \mathbb{C},}$ then ${\displaystyle V_{\mathbb{R}}={\mathbb{R}}^2}$ is a vector space over ${\displaystyle \mathbb{R}}$ and ${\displaystyle ⟨x,y{⟩}_{\mathbb{R}}}$ is the dot product ${\displaystyle x\cdot y,}$ where ${\displaystyle x=a+ib\in V=\mathbb{C}}$ is identified with the point ${\displaystyle (a,b)\in V_{\mathbb{R}}={\mathbb{R}}^2}$ (and similarly for ${\displaystyle y}$); thus the standard inner product ${\displaystyle ⟨x,y⟩=x\overline{y},}$ on ${\displaystyle \mathbb{C}}$ is an "extension" the dot product . Also, had ${\displaystyle ⟨x,y⟩}$ been instead defined to be the symmetric map ${\displaystyle ⟨x,y⟩=xy}$ (rather than the usual conjugate symmetric map ${\displaystyle ⟨x,y⟩=x\overline{y}}$) then its real part ${\displaystyle ⟨x,y{⟩}_{\mathbb{R}}}$ would not be the dot product; furthermore, without the complex conjugate, if ${\displaystyle x\in \mathbb{C}}$ but ${\displaystyle x\notin \mathbb{R}}$ then ${\displaystyle ⟨x,x⟩=xx=x^2\notin [0,\mathrm{\infty })}$ so the assignment ${\displaystyle x\mapsto \sqrt{⟨x,x⟩}}$ would not define a norm.

The next examples show that although real and complex inner products have many properties and results in common, they are not entirely interchangeable. For instance, if ${\displaystyle ⟨x,y⟩=0}$ then ${\displaystyle ⟨x,y{⟩}_{\mathbb{R}}=0,}$ but the next example shows that the converse is in general not true. Given any ${\displaystyle x\in V,}$ the vector ${\displaystyle ix}$ (which is the vector ${\displaystyle x}$ rotated by 90°) belongs to ${\displaystyle V}$ and so also belongs to ${\displaystyle V_{\mathbb{R}}}$ (although scalar multiplication of ${\displaystyle x}$ by ${\displaystyle i=\sqrt{-1}}$ is not defined in ${\displaystyle V_{\mathbb{R}},}$ the vector in ${\displaystyle V}$ denoted by ${\displaystyle ix}$ is nevertheless still also an element of ${\displaystyle V_{\mathbb{R}}}$). For the complex inner product, ${\displaystyle ⟨x,ix⟩=-i\Vert x{\Vert }^2,}$ whereas for the real inner product the value is always ${\displaystyle ⟨x,ix{⟩}_{\mathbb{R}}=0.}$

If ${\displaystyle ⟨\cdot ,\cdot ⟩}$ is a complex inner product and ${\displaystyle A:V\to V}$ is a continuous linear operator that satisfies ${\displaystyle ⟨x,Ax⟩=0}$ for all ${\displaystyle x\in V,}$ then ${\displaystyle A=0.}$ This statement is no longer true if ${\displaystyle ⟨\cdot ,\cdot ⟩}$ is instead a real inner product, as this next example shows. Suppose that ${\displaystyle V=\mathbb{C}}$ has the inner product ${\displaystyle ⟨x,y⟩:=x\overline{y}}$ mentioned above. Then the map ${\displaystyle A:V\to V}$ defined by ${\displaystyle Ax=ix}$ is a linear map (linear for both ${\displaystyle V}$ and ${\displaystyle V_{\mathbb{R}}}$) that denotes rotation by ${\displaystyle {90}^{\circ }}$ in the plane. Because ${\displaystyle x}$ and ${\displaystyle Ax}$ are perpendicular vectors and ${\displaystyle ⟨x,Ax{⟩}_{\mathbb{R}}}$ is just the dot product, ${\displaystyle ⟨x,Ax{⟩}_{\mathbb{R}}=0}$ for all vectors ${\displaystyle x;}$ nevertheless, this rotation map ${\displaystyle A}$ is certainly not identically ${\displaystyle 0.}$ In contrast, using the complex inner product gives ${\displaystyle ⟨x,Ax⟩=-i\Vert x{\Vert }^2,}$ which (as expected) is not identically zero.

Orthonormal sequences

See also: Orthogonal basis and Orthonormal basis

Let ${\displaystyle V}$ be a finite dimensional inner product space of dimension ${\displaystyle n.}$ Recall that every basis of ${\displaystyle V}$ consists of exactly ${\displaystyle n}$ linearly independent vectors. Using the Gram–Schmidt process we may start with an arbitrary basis and transform it into an orthonormal basis. That is, into a basis in which all the elements are orthogonal and have unit norm. In symbols, a basis ${\displaystyle \{e_1,\dots ,e_n\}}$ is orthonormal if ${\displaystyle ⟨e_i,e_j⟩=0}$ for every ${\displaystyle i\ne j}$ and ${\displaystyle ⟨e_i,e_i⟩=\Vert e_a{\Vert }^2=1}$ for each index ${\displaystyle i.}$

This definition of orthonormal basis generalizes to the case of infinite-dimensional inner product spaces in the following way. Let ${\displaystyle V}$ be any inner product space. Then a collection $${\displaystyle E={\left\{e_a\right\}}_{a\in A}}$$ is a basis for ${\displaystyle V}$ if the subspace of ${\displaystyle V}$ generated by finite linear combinations of elements of ${\displaystyle E}$ is dense in ${\displaystyle V}$ (in the norm induced by the inner product). Say that ${\displaystyle E}$ is an orthonormal basis for ${\displaystyle V}$ if it is a basis and $${\displaystyle ⟨e_a,e_b⟩=0}$$ if ${\displaystyle a\ne b}$ and ${\displaystyle ⟨e_a,e_a⟩=\Vert e_a{\Vert }^2=1}$ for all ${\displaystyle a,b\in A.}$

Using an infinite-dimensional analog of the Gram-Schmidt process one may show:

Theorem. Any separable inner product space has an orthonormal basis.

Using the Hausdorff maximal principle and the fact that in a complete inner product space orthogonal projection onto linear subspaces is well-defined, one may also show that

Theorem. Any complete inner product space has an orthonormal basis.

The two previous theorems raise the question of whether all inner product spaces have an orthonormal basis. The answer, it turns out is negative. This is a non-trivial result, and is proved below. The following proof is taken from Halmos's A Hilbert Space Problem Book (see the references).

Proof

Parseval's identity leads immediately to the following theorem:

Theorem. Let ${\displaystyle V}$ be a separable inner product space and ${\displaystyle {\left\{e_k\right\}}_k}$ an orthonormal basis of ${\displaystyle V.}$ Then the map $${\displaystyle x\mapsto \{⟨e_k,x⟩{\}}_{k\in \mathbb{N}}}$$ is an isometric linear map ${\displaystyle V\to {\ell }^2}$ with a dense image.

This theorem can be regarded as an abstract form of Fourier series, in which an arbitrary orthonormal basis plays the role of the sequence of trigonometric polynomials. Note that the underlying index set can be taken to be any countable set (and in fact any set whatsoever, provided ${\displaystyle {\ell }^2}$ is defined appropriately, as is explained in the article Hilbert space). In particular, we obtain the following result in the theory of Fourier series:

Theorem. Let ${\displaystyle V}$ be the inner product space ${\displaystyle C[-\pi ,\pi ].}$ Then the sequence (indexed on set of all integers) of continuous functions $${\displaystyle e_k(t)=\frac{e^{ikt}}{\sqrt{2\pi }}}$$ is an orthonormal basis of the space ${\displaystyle C[-\pi ,\pi ]}$ with the ${\displaystyle L^2}$ inner product. The mapping $${\displaystyle f\mapsto \frac{1}{\sqrt{2\pi }}{\left\{{\int }_{-\pi }^{\pi }f(t)e^{-ikt}\mathrm{d}t\right\}}_{k\in \mathbb{Z}}}$$ is an isometric linear map with dense image.

Orthogonality of the sequence ${\displaystyle \{e_k{\}}_k}$ follows immediately from the fact that if ${\displaystyle k\ne j,}$ then $${\displaystyle {\int }_{-\pi }^{\pi }{e}^{-i(j-k)t}\mathrm{d}t=0.}$$

Normality of the sequence is by design, that is, the coefficients are so chosen so that the norm comes out to 1. Finally the fact that the sequence has a dense algebraic span, in the inner product norm, follows from the fact that the sequence has a dense algebraic span, this time in the space of continuous periodic functions on ${\displaystyle [-\pi ,\pi ]}$ with the uniform norm. This is the content of the Weierstrass theorem on the uniform density of trigonometric polynomials.

Operators on inner product spaces

Main article: Operator theory

Several types of linear maps ${\displaystyle A:V\to W}$ between inner product spaces ${\displaystyle V}$ and ${\displaystyle W}$ are of relevance:

• Continuous linear maps: ${\displaystyle A:V\to W}$ is linear and continuous with respect to the metric defined above, or equivalently, ${\displaystyle A}$ is linear and the set of non-negative reals ${\displaystyle \{\Vert Ax\Vert :\Vert x\Vert \le 1\},}$ where ${\displaystyle x}$ ranges over the closed unit ball of ${\displaystyle V,}$ is bounded.
• Symmetric linear operators: ${\displaystyle A:V\to W}$ is linear and ${\displaystyle ⟨Ax,y⟩=⟨x,Ay⟩}$ for all ${\displaystyle x,y\in V.}$
• Isometries: ${\displaystyle A:V\to W}$ satisfies ${\displaystyle \Vert Ax\Vert =\Vert x\Vert }$ for all ${\displaystyle x\in V.}$ A linear isometry (resp. an antilinear isometry) is an isometry that is also a linear map (resp. an antilinear map). For inner product spaces, the polarization identity can be used to show that ${\displaystyle A}$ is an isometry if and only if ${\displaystyle ⟨Ax,Ay⟩=⟨x,y⟩}$ for all ${\displaystyle x,y\in V.}$ All isometries are injective. The Mazur–Ulam theorem establishes that every surjective isometry between two real normed spaces is an affine transformation. Consequently, an isometry ${\displaystyle A}$ between real inner product spaces is a linear map if and only if ${\displaystyle A(0)=0.}$ Isometries are morphisms between inner product spaces, and morphisms of real inner product spaces are orthogonal transformations (compare with orthogonal matrix).
• Isometrical isomorphisms: ${\displaystyle A:V\to W}$ is an isometry which is surjective (and hence bijective). Isometrical isomorphisms are also known as unitary operators (compare with unitary matrix).

From the point of view of inner product space theory, there is no need to distinguish between two spaces which are isometrically isomorphic. The spectral theorem provides a canonical form for symmetric, unitary and more generally normal operators on finite dimensional inner product spaces. A generalization of the spectral theorem holds for continuous normal operators in Hilbert spaces.

Generalizations

Any of the axioms of an inner product may be weakened, yielding generalized notions. The generalizations that are closest to inner products occur where bilinearity and conjugate symmetry are retained, but positive-definiteness is weakened.

Degenerate inner products

Main article: Krein space

If ${\displaystyle V}$ is a vector space and ${\displaystyle ⟨\cdot ,\cdot ⟩}$ a semi-definite sesquilinear form, then the function: $${\displaystyle \Vert x\Vert =\sqrt{⟨x,x⟩}}$$ makes sense and satisfies all the properties of norm except that ${\displaystyle \Vert x\Vert =0}$ does not imply ${\displaystyle x=0}$ (such a functional is then called a semi-norm). We can produce an inner product space by considering the quotient ${\displaystyle W=V/\{x:\Vert x\Vert =0\}.}$ The sesquilinear form ${\displaystyle ⟨\cdot ,\cdot ⟩}$ factors through ${\displaystyle W.}$

This construction is used in numerous contexts. The Gelfand–Naimark–Segal construction is a particularly important example of the use of this technique. Another example is the representation of semi-definite kernels on arbitrary sets.

Nondegenerate conjugate symmetric forms

Main article: Pseudo-Euclidean space

Alternatively, one may require that the pairing be a nondegenerate form, meaning that for all non-zero ${\displaystyle x\ne 0}$ there exists some ${\displaystyle y}$ such that ${\displaystyle ⟨x,y⟩\ne 0,}$ though ${\displaystyle y}$ need not equal ${\displaystyle x}$; in other words, the induced map to the dual space ${\displaystyle V\to V^{\ast }}$ is injective. This generalization is important in differential geometry: a manifold whose tangent spaces have an inner product is a Riemannian manifold, while if this is related to nondegenerate conjugate symmetric form the manifold is a pseudo-Riemannian manifold. By Sylvester's law of inertia, just as every inner product is similar to the dot product with positive weights on a set of vectors, every nondegenerate conjugate symmetric form is similar to the dot product with nonzero weights on a set of vectors, and the number of positive and negative weights are called respectively the positive index and negative index. Product of vectors in Minkowski space is an example of indefinite inner product, although, technically speaking, it is not an inner product according to the standard definition above. Minkowski space has four dimensions and indices 3 and 1 (assignment of "+" and "−" to them differs depending on conventions).

Purely algebraic statements (ones that do not use positivity) usually only rely on the nondegeneracy (the injective homomorphism ${\displaystyle V\to V^{\ast }}$) and thus hold more generally.

Related products

The term "inner product" is opposed to outer product, which is a slightly more general opposite. Simply, in coordinates, the inner product is the product of a ${\displaystyle 1\times n}$ covector with an ${\displaystyle n\times 1}$ vector, yielding a ${\displaystyle 1\times 1}$ matrix (a scalar), while the outer product is the product of an ${\displaystyle m\times 1}$ vector with a ${\displaystyle 1\times n}$ covector, yielding an ${\displaystyle m\times n}$ matrix. The outer product is defined for different dimensions, while the inner product requires the same dimension. If the dimensions are the same, then the inner product is the trace of the outer product (trace only being properly defined for square matrices). In an informal summary: "inner is horizontal times vertical and shrinks down, outer is vertical times horizontal and expands out".

More abstractly, the outer product is the bilinear map ${\displaystyle W\times V^{\ast }\to \mathrm{hom}(V,W)}$ sending a vector and a covector to a rank 1 linear transformation (simple tensor of type (1, 1)), while the inner product is the bilinear evaluation map ${\displaystyle V^{\ast }\times V\to F}$ given by evaluating a covector on a vector; the order of the domain vector spaces here reflects the covector/vector distinction.

The inner product and outer product should not be confused with the interior product and exterior product, which are instead operations on vector fields and differential forms, or more generally on the exterior algebra.

As a further complication, in geometric algebra the inner product and the exterior (Grassmann) product are combined in the geometric product (the Clifford product in a Clifford algebra) – the inner product sends two vectors (1-vectors) to a scalar (a 0-vector), while the exterior product sends two vectors to a bivector (2-vector) – and in this context the exterior product is usually called the outer product (alternatively, wedge product). The inner product is more correctly called a scalar product in this context, as the nondegenerate quadratic form in question need not be positive definite (need not be an inner product).