# Functional analysis

## Functional analysis Topics

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### Operator spectrum

Let be a linear operator on a separable Hilbert space. The spectrum of is the set of such that is not invertible on all of the Hilbert space, where the s are complex numbers and is the identity operator. The definition can also be stated in terms of the resolvent of an operatorand then the spectrum is defined to be the complement of in the complex plane. It is easy to demonstrate that is an open set, which shows that the spectrum is closed.If is a domain in (i.e., a Lebesgue measurable subset of with finite nonzero Lebesgue measure), then a set is a spectrum of if is an orthogonal basis of (Iosevich et al. 1999).

### Norm topology

The norm topology on a normed space is the topology consisting of all sets which can be written as a (possibly empty) union of sets of the formfor some and for some number . Sets of the form are called the open balls in .

### Wiener algebra

Suppose is the set of all complex-valued functions on the interval of the form(1)for , where the and . The set with the usual pointwise operations and with the norm(2)is a commutative Banach algebra and is called theWiener algebra.There is an isometric isomorphism between and given by , where(3)with .

### Nonmeager set

A subset of a topological space is said to be nonmeager if is of second category in , i.e., if cannot be written as the countable union of subsets which are nowhere dense in .

### Weighted shift

Let , be a bounded sequence of complex numbers, and be the (usual) standard orthonormal basis of , that is, , , where denoted the Kronecker delta, so thatfor any . Then the operator defined by is called a weighted shift with the weights . Then ,and and .If for all , then is called unilateral shift operator.

### Nondegenerate operator action

A -algebra of operators on a Hilbert space is said to act nondegenerately if whenever for all , it necessarily implies that . Algebras which act nondegenerately are sometimes said to be nondegenerate.One can show that such an algebra is nondegenerate if and only if the subspaceis dense in .Any -algebra containing the identity operator necessarily acts nondegenerately.

### Weakly complemented subspace

A closed subspace of a Banach space is called weakly complemented if the dual of the natural embedding has a right inverse as a bounded operator.For example, the Banach space of all complex sequences converging to zero together with the supremum norm is weakly complemented in , not complemented in (Whitley 1966).

### Noncommutative topology

Noncommutative topology is a recent program having important and deep applications in several branches of mathematics and mathematical physics. Because every commutative -algebra is -isomorphic to where is the space of maximal ideals of (this is the so-called Gelfand theorem) and because an algebraic isomorphism between and induces a homeomorphism between and , -algebraic theory may be regarded as a noncommutative analogue of the algebra of continuous functions vanishing at infinity on a locally compact T2-space. In other words, every property of a locally compact T2-space can be formulated in terms of a "Gelfand dual" property of and then it will probably be true for any noncommutative -algebra.The following is a list of some such Gelfand dualities (Wegge-Olsen 1993, Moslehian 2002):topological language-algebraic languagelocally compact T2-spaceC-*-algebratriangulation or the structure of an affine algebraic..

### Extreme set

Given a subset of a vector space , a nonempty subset is called an extreme set of if no point of is an internal point of any line interval whose endpoints are in except when both endpoints are in . Said another way, is an extreme set of if whenever andfor , it necessarily follows that .In the event that consists of a single point of , is called an extreme point of . Extreme points play an important role in a number of areas of math, e.g., in the Krein-Milman theorem in functional analysis.

### Weakly amenable

The notion of weak amenability was first introduced by Bade et al. (1987), who termed a commutative Banach algebra "weakly amenable" if every continuous derivation from into a symmetric Banach -bimodule is zero. But this is equivalent to ,and one may apply this latter condition as the definition of weak amenability for an arbitrary Banach algebra. So a Banach algebra is said to be weakly amenable if every bounded derivation from into its dual is inner (Helemskii 1989).It is known that every -algebra is weakly amenable (Haagerup 1983).

### Nest and nest algebra

Let be a complex Hilbert space, and define a nest as a set of closed subspaces of satisfying the conditions:1. , 2. If , then either or , 3. If ,then , 4. If ,then the norm closure of the linear span of lies in . (Davidson 1988).The nest algebra associated with the nest is the set .For example, consider an orthonormal basis of a separable Hilbert space . Put . Then is a nest and the associated nest algebra is the algebra of operators whose matrix representation with respect to is upper triangular.

### Extreme point

An extreme point of a subset of a vector space is an extreme set of which consists of a single point in . The collection of all extreme points of is sometimes denoted .Extreme points play important roles in a number of areas of mathematics, e.g., in the Krein-Milman theorem which says that, despite their moniker implying a certain degree of rarity, the collection can be quite large relative the host space .

### Weak topology

Let be a topological vector space whose continuous dual separates points (i.e., is T2). The weak topology on is defined to be the coarsest/weakest topology (that is, the topology with the fewest open sets) under which each element of remains continuous on . To differentiate the topologies and , is sometimes referred to as the strong topology on .Note that the weak topology is a special case of a more general concept. In particular, given a nonempty family of mappings from a set to a topological space , one can define a topology to be the collection of all unions and finite intersections of sets of the form with and an open set in . The topology -often called the -topology on and/or the weak topology on induced by -is the coarsest topology in which every element is continuous on and so it follows that the above-stated definition corresponds to the special case of for a topological vector space...

### Equicontinuous

In real and functional analysis, equicontinuity is a concept which extends the notion of uniform continuity from a single function to collection of functions. Given topological vector spaces and , a collection of linear transformations from into is said to be equicontinuous if to every neighborhood of in there corresponds a neighborhood of in such that for all . In the special case that is a metric space and , this criterion can be restated as an epsilon-delta definition: A collection of real-valued continuous functions on is equicontinuous if, given , there is a such that whenever satisfy ,for all . It is often convenient to visualize an equicontinuous collection of functions as being "uniformly uniformly continuous," i.e., a collection for which a single can be chosen for any arbitrary so as to make all uniformly continuous simultaneously, independent of .In the latter case, equicontinuity is the ingredient needed to "upgrade"..

### Von neumann algebra

Given a Hilbert space , a -subalgebra of is said to be a von Neumann algebra in provided that is equal to its bicommutant (Dixmier 1981). Here, denotes the algebra of bounded operators from to itself.A non-trivial corollary of the so-called bicommutant theorem says that a nondegenerate -subalgebra of is a von Neumann algebra if and only if it is strongly closed. This is further equivalent to a number of other analytic properties of and of (Blackadar 2013), and due to its bijective equivalence is sometimes used as a definition for von Neumann algebras. In some literature, the assumption of being unital (i.e., containing the identity) is added to the hypotheses of this equivalence though, strictly speaking, the result holds in the somewhat more general case that is merely nondegenerate.One can easily show that every von Neumann algebra is a W-*-algebra and contrarily; as a result, some literature defines a von Neumann algebra as a C-*-algebra which..

### Eigenfunction

If is a linear operator on a function space, then is an eigenfunction for and is the associated eigenvalue whenever .Renteln and Dundes (2005) give the following (bad) mathematical joke about eigenfunctions:Q: What do you call a young eigensheep? A: A lamb, duh!

### Dual vector space

The dual vector space to a real vector space is the vector space of linear functions , denoted . In the dual of a complex vector space, the linear functions take complex values.In either case, the dual vector space has the same dimension as . Given a vector basis , ..., for there exists a dual basis for , written , ..., , where and is the Kronecker delta.Another way to realize an isomorphism with is through an inner product. A real vector space can have a symmetric inner product in which case a vector corresponds to a dual element by . Then a basis corresponds to its dual basis only if it is an orthonormal basis, in which case . A complex vector space can have a Hermitian inner product, in which case is a conjugate-linear isomorphism of with , i.e., .Dual vector spaces can describe many objects in linear algebra. When and are finite dimensional vector spaces, an element of the tensor product , say , corresponds to the linear transformation . That is, . For example,..

### Vector space polar

Given a topological vector space and a neighborhood of in , the polar of is defined to be the setand where denotes the magnitude of the scalar in the underlying scalar field of (i.e., the absolute value of if is a real vector space or its complex modulus if is a complex vector space) and where denotes the continuous dual space of (i.e., is the space of all continuous linear functionals from to the underlying scalar field of ).Worth noting is that the polar is essentially the norm unit ball in and is fundamental in functional analysis, e.g., in the Banach-Alaoglu theorem which says is weak-* compact for all neighborhoods of in .

### Disk algebra

A disk algebra is an algebra of functions which are analytic on the open unit disk in and continuous up to the boundary. A representative measure for a point in the closed disk is a nonnegative measure such that for all in . These measures form a compact, convex set in the linear space of all measures.Stated another way, let denote the closed unit disk . Suppose that denoted the set of all elements of which are analytic on the interior of . is a closed subalgebra of and so is a unital commutative Banach algebra. This algebra is called the disk algebra.

### Discrete semigroup algebra

Let be a semigroup and a positive real-valued function on such that   . If is the set of all complex-valued functions on for which , then with the usual pointwise addition, scalar multiplication, the product (convolution) (if has no solutions, we assume ), and with the norm is a Banach algebra.If , then is called discrete semi-group algebra. Moreover if is a group then is the discrete group algebra .

### Discrete group algebra

Let be an algebraic group. together with the discrete topology is a locally compact group and one may consider the counting measure as a left invariant Haar measure on . Then denoted and is called a discrete group algebra.

### Minimal banach space

A Banach space is called minimal if every infinite-dimensional subspace of contains a subspace isomorphic to . An example of a minimal Banach space is the Banach space of all complex sequences converging to zero (taking the supremum norm).

### Dirichlet's principle

Dirichlet's principle, also known as Thomson's principle, states that there exists a function that minimizes the functional(called the Dirichlet integral) for or among all the functions which take on given values on the boundary of , and that function satisfies in , , . Weierstrass showed that Dirichlet's argument contained a subtle fallacy. As a result, it can be claimed only that there exists a lower bound to which comes arbitrarily close without being forced to actually reach it. Kneser, however, obtained a valid proof of Dirichlet's principle.

### Milman's theorem

Let be a locally convex topological vector space and let be a compact subset of . In functional analysis, Milman's theorem is a result which says that if the closed convex hull of is also compact, then contains all the extreme points of .The importance of Milman's theorem is subtle but enormous. One well-known fact from functional analysis is that where denotes the set of extreme points of . Ostensibly, however, one may have that has extreme points which are not in . This behavior is considered a pathology, and Milman's theorem states that this pathology cannot exist whenever is compact (e.g., when is a subset of a Fréchet space ).Milman's theorem should not be confused with the Krein-Milman theorem which says that every nonempty compact convex set in necessarily satisfies the identity ...

### Derivation

A derivation is a sequence of steps, logical or computational, from one result to another. The word derivation comes from the word "derive.""Derivation" can also refer to a particular type of operator used to define a derivation algebra on a ring or algebra. In particular, let be a Banach algebra and be a Banach -bimodule. Any element ofis called a bounded derivation of in and any element ofis called an inner derivation.

### Uniformly convex

A normed vector space is said to be uniformly convex if for sequences , , the assumptions , , and together imply thatas tends to infinity.Such spaces are important in functional analysis. For example, the classical Banach-Saks theorem can be generalized so that the desired conclusion holds in the case that is a Banach space whose conjugate space (that is, the complex conjugate of the dual vector space ) is uniformly convex.

### Meager set

A subset of a topological space is said to be meager if is of first category in , i.e., if can be written as the countable union of subsets which are nowhere dense in . The terms "thin set," "meager set," and "first category" are equivalent.

### Uniformly continuous

A map from a metric space to a metric space is said to be uniformly continuous if for every , there exists a such that whenever satisfy .Note that the here depends on and on but that it is entirely independent of the points and . In this way, uniform continuity is stronger than continuity and so it follows immediately that every uniformly continuous function is continuous.Examples of uniformly continuous functions include Lipschitz functions and those satisfying the Hölder condition. Note however that not all continuous functions are uniformly continuous with two very basic counterexamples being (for ) and (for . On the other hand, every function which is continuous on a compact domain is necessarily uniformly continuous.

### Maximal ideal space

Let be a commutative complex Banach algebra. The space of all characters on is called the maximal ideal space (or character space) of . This space equipped with the weak*-topology inherited from is a locally compact T2-space.

### Uniform boundedness principle

A "pointwise-bounded" family of continuous linear operators from a Banach space to a normed space is "uniformly bounded." Symbolically, if is finite for each in the unit ball, then is finite. The theorem is a corollary of the Banach-Steinhaus theorem.Stated another way, let be a Banach space and be a normed space. If is a collection of bounded linear mappings of into such that for each , then .

### Convergence in mean

The phrase "convergence in mean" is used in several branches of mathematics to refer to a number of different types of sequential convergence.In functional analysis, "convergence in mean" is most often used as another name for strong convergence. In particular, a sequence in a normed linear space converges in mean to an element wheneveras , where denotes the norm on . Sometimes, however, a sequence of functions in is said to converge in mean if converges in -norm to a function for some measure space .The term is also used in probability and related theories to mean something somewhat different. In these contexts, a sequence of random variables is said to converge in the th mean (or in the norm) to a random variable if the th absolute moments and all exist and ifwhere denotes the expectation value. In this usage, convergence in the norm for the special case is called "convergence in mean."..

### Trace operator

Let be a bounded open set in whose boundary is at least smooth and let(1)be a linear operator defined by(2)on the collection of all real-valued compactly-supported functions with domain in the topological closure of . In functional analysis, the trace operator is defined to be the extension(3)of to functions whose domain is the Sobolev space . Intuitively, the trace operator literally "traces" the boundary of a function . This piece of data is of particular important when studying function spaces and partial differential equations due to the existence of various boundary-value parameters in these contexts.

### Locally convex

A topology on a topological vector space (with usually assumed to be T2) is said to be locally convex if admits a local base at consisting of balanced, convex, and absorbing sets. In some older literature, the definition of locally convex is often stated without requiring that the local base be balanced or absorbing.It is not unusual to blur the distinction as to whether "locally convex" applies to the topology on or to itself.The above definition can also be stated in terms of seminorms. In particular, a topological vector space (with assumed ) is locally convex if is generated by a family of seminorms satisfyingwhere denotes the zero vector in and is different from 0 which denotes the element 0 in the scalar field of . The condition (1) above ensures that is ; removal of this criterion on allows one to remove condition (1), whereby is locally convex if and only if is generated by a family of seminorms.The seminorm condition illustrates why..

### Contractible banach algebra

A Banach algebra is called contractible if for all Banach -bimodules (Helemskii 1989, 1997).A -algebra is contractible if and only if it is finite-dimensional (Selivanov 1976).

### Local base

Let be a topological vector space and for an arbitrary point , denote by the collection of all neighborhoods of in . A local base at is any set for which each element includes some member of .A local base at a point is sometimes called a local base of neighborhoods at that point.

### Local banach algebra

A local Banach algebra is a normed algebra which satisfies the following properties: 1. If and is an analytic function on a neighborhood of the spectrum of in the completion of , with if is non-unital, then . 2. All matrix algebras over satisfy property (1) above. Here, if is a *-algebra, then it will be called a local Banach -algebra; similarly, if is a pre--norm, then is called a local C-*-algebra (though different literature uses the term "local -algebra" to refer to different structures altogether).An algebra satisfying condition (1) above is said to be closed under holomorphic functional calculus.

### Concentrated

Let be a positive measure on a sigma-algebra , and let be an arbitrary (real or complex) measure on . If there is a set such that for every , then is said to be concentrated on . This is equivalent to requiring that whenever .

### Stampacchia theorem

"Stampacchia's theorem" is a name given to any number of related results in functional analysis, and while the body of the theorem often varies depending on the literature consulted, one commonly-encountered result attributed to Stampacchia is a sort of "representation inequality" for continuous, coercive bilinear forms on an arbitrary Hilbert space . This particular version of the result is considerable for a number of reasons, most notably for its implication of the so-called Lax-Milgram theorem.To state the above-mentioned version of the theorem, let be a Hilbert space, let be a continuous and coercive bilinear form on , and let be a closed and convex subset of . One result of Stampacchia says that, under these assumptions, any function necessarily corresponds to a unique function for which the inequality(1)is satisfied for all functions where here, denotes the inner product on . Note that this form of the result..

### Linear space

There are at least two distinct notions of linear space throughout mathematics.The term linear space is most commonly used within functionalanalysis as a synonym of the term vector space.The term is also used to describe a fundamental notion in the field of incidence geometry. In particular, a linear space is a space consisting of a collection of points and a set of lines subject to the following axioms: 1. Any two distinct points of belong to exactly one line of . 2. Any line of has at least two points of . 3. There are at least three points of not on a common line. Using this terminology, lines are considered to be "distinguished subsets" of the collection of points. Moreover, in this context, one can view a linear space as a generalization of the notions of projective space and affine space (Batten and Beutelspracher 2009)...

### Complemented subspace

Let be a normed space, and be algebraically complemented subspaces of (i.e., and ), be the quotient map, be the natural isomorphism , and be the projection of on along . Then the following statements are equivalent:1. is a homeomorphism. 2. and are closed in and is a homeomorphism. 3. and are closed and is a bounded projection. The subspaces and are called topologically complemented or simply complemented if each of the above equivalent statements holds (Constantinescu 2001, Meise and Vogt 1997).Every finite dimensional subspace is complemented and every algebraic complement of a finite codimension subspace is topologically complemented. In a Banach space , two closed subspace are algebraically complemented if and only if they are complemented.There are uncomplemented closed subspaces. For example, let be the disk algebra, i.e., the space of all analytic functions on which are continuous on the closure of . Then the subspace of consisting..

### Linear functional

A linear functional on a real vector space is a function , which satisfies the following properties. 1. , and 2. . When is a complex vector space, then is a linear map into the complex numbers.Generalized functions are a special caseof linear functionals, and have a rich theory surrounding them.

### Complementary subspace problem

The complementary subspace problem asks, in general, which closed subspaces of a Banach space are complemented (Johnson and Lindenstrauss 2001).Phillips (1940) proved that the Banach space of all complex sequences converging to zero together with the supremum norm is uncomplemented in the L-infinity-space of positive integers .Pełczyński (1960) showed that complemented subspaces of , the Banach space of all absolutely summable complex sequences equipped with -norm, are isomorphic to .In 1971, Lindenstrauss and Tzafriri (1977) proved that every infinite-dimensional Banach space that is not isomorphic to a Hilbert space contains a closed uncomplemented subspace.Pisier (1992) established that any complemented reflexive subspace of a -algebra is necessarily linearly isomorphic to a Hilbert space.Gowers and Maurey (1993) showed that there exists a Banach space without nontrivial complemented subspaces...

### Compact operator

If and are Banach spaces and is a bounded linear operator, the is said to be a compact operator if it maps the unit ball of into a relatively compact subset of (that is, a subset of with compact closure).The basic example of a compact operator is an infinite diagonal matrix with . The matrix gives a bounded map , where is the set of square-integrable sequences. It is a compact operator because it is the limit of the finite rank matrices , which have the same entries as except for . That is, the have only finitely many nonzero entries.The properties of compact operators are similar to those of finite-dimensional linear transformations. For Hilbert spaces, any compact operator is the limit of a sequence of operators with finite rank, i.e., the image of is a finite-dimensional subspace in . However, this property does not hold in general as shown by Enflo (1973), who constructed a Banach space that provides a counterexample, thus solving the approximation..

### Sobolev embedding theorem

The Sobolev embedding theorem is a result in functional analysis which proves that certain Sobolev spaces can be embedded in various spaces including , , and for various domains , in and for miscellaneous values of , , , , , , and (usually depending on properties of the domains and ). Because numerous such embeddings are possible, many individual results may be termed "the" Sobolev embedding theorem, whereas in actuality the phrase "Sobolev embedding theorem" is best thought of as an umbrella term encompassing all such results.To proceed, let be a domain (i.e., a bounded, connected open set) in and let be the intersection of with a hyperplane of dimension in for . Let , be integers and let . Under these constructions, one has a number of function space embeddings, the collection of which will be referred to as the Sobolev embedding theorem.For example, if satisfies a so-called "cone condition" (i.e., if there exists..

### Commutant

Given a complex Hilbert space with associated space of continuous linear operators from to itself, the commutant of an arbitrary subset is the collection of all elements in which commute with all elements of : .Across the literature on the subject, the set is sometimes denoted , a reference to the fact that a linear operator between normed vector spaces is continuous if and only if it is bounded (Royden and Fitzpatrick 2010).The notions of commutant and bicommutant are fundamental to the study of von Neumann algebras (Dixmier 1981).

### Coercive functional

A bilinear functional on a normed space is called coercive (or sometimes elliptic) if there exists a positive constant such thatfor all .

### Shift transformation

The transformation(1)(2)where is the fractional part of and is the floor function, that takes a continued fraction to .

### Seminorm

A seminorm is a function on a vector space , denoted , such that the following conditions hold for all and in , and any scalar . 1. , 2. , and 3. . Note that it is possible for for nonzero . For example, the functional for continuous functions is a seminorm which is not a norm. A seminorm is a norm if is equivalent to .

### Second category

A subset of a topological space is said to be of second category in if cannot be written as the countable union of subsets which are nowhere dense in , i.e., if writing as a unionimplies that at least one subset fails to be nowhere dense in . Said differently, any set which fails to be of first category is necessarily second category and unlike sets of first category, one thinks of a second category subset as a "non-small" subset of its host space. Sets of second category are sometimes referred to as nonmeager.An important distinction should be made between the above-used notion of "category" and category theory. Indeed, the notions of first and second category sets are independent of category theory.The irrational numbers are of second category and the rational numbers are of first category in with the usual topology. In general, the host space and its topology play a fundamental role in determining category. For example, the..

### Ces&agrave;ro mean

The Cesàro means of a function are the arithmetic means(1), 2, ..., where the addend is the th partial sum(2)of the Fourier series(3)for . Here, is the th coefficient(4)in the Fourier expansion for , . Cesàro means are of particular importance in the study of function spaces. For example, a well-known fact is that if is a -integrable function for , the Cesàro means of converge to in the -norm and, moreover, if is continuous, the convergence is uniform. The th Cesàro mean of can also be obtained by integrating against the th Fejer kernel.

### Cauchy functional equation

Cauchy's functional equation is the equationIt was proved by Cauchy in 1821 that the only continuous solutions of this functional equation from into are those of the form for some real number . In 1875, Darboux showed that the continuity hypothesis could be replaced by continuity at a single point and, five years later, proved that it would be enough to assume that is nonnegative (or nonpositive) for sufficiently small positive .In 1905, G. Hamel proved that there are non-continuous solutions of the Cauchy functional equation using Hamel bases. Every non-continuous solution is necessarily non-measurable with respect to the Lebesgue measure.The fifth of Hilbert's problems is a generalizationof this equation.

### Schauder basis

A Schauder basis for a Banach space is a sequence in with the property that every has a unique representation of the form for in which the sum is convergent in the norm topology. For example, the trigonometrical system is a basis in each space for .

### Kakutani's fixed point theorem

Kakutani's fixed point theorem is a result in functional analysis which establishes the existence of a common fixed point among a collection of maps defined on certain "well-behaved" subsets of locally convex topological vector spaces. The theorem is relevant both because of its independent theoretical significance and because of other results which stem as corollaries therefrom.One common form of Kakutani's fixed point theorem states that, given a locally convex topological vector space , any equicontinuous group of affine maps mapping a (nonempty) compact, convex subset into itself necessarily has a common fixed point in , i.e., that under the above conditions, there exists a point satisfying for all . In addition, Markov and Kakutani proved that some of these hypotheses can be weakened without affecting the results, e.g., that the result remains true for arbitrary topological vector spaces (which may or may not be locally..

### Riesz's theorem

Every continuous linear functional for can be expressed as a Stieltjes integralwhere is determined by and is of bounded variation on .

### Inner product

An inner product is a generalization of the dot product. In a vector space, it is a way to multiply vectors together, with the result of this multiplication being a scalar.More precisely, for a real vector space, an inner product satisfies the following four properties. Let , , and be vectors and be a scalar, then: 1. . 2. . 3. . 4. and equal if and only if . The fourth condition in the list above is known as the positive-definite condition. Related thereto, note that some authors define an inner product to be a function satisfying only the first three of the above conditions with the added (weaker) condition of being (weakly) non-degenerate (i.e., if for all , then ). In such literature, functions satisfying all four such conditions are typically referred to as positive-definite inner products (Ratcliffe 2006), though inner products which fail to be positive-definite are sometimes called indefinite to avoid confusion. This difference, though subtle,..

### Bump function

Given any open set in with compact closure , there exist smooth functions which are identically one on and vanish arbitrarily close to . One way to express this more precisely is that for any open set containing , there is a smooth function such that 1. for all and 2. for all . A function that satisfies (1) and (2) is called a bump function. If then by rescaling , namely , one gets a sequence of smooth functions which converges to the delta function, providing that is a neighborhood of 0.

### Bounded operator

A bounded operator between two Banach spaces satisfies the inequality(1)where is a constant independent of the choice of . The inequality is called a bound. For example, consider , which has L2-norm . Then is a bounded operator,(2)from L2-space to L1-space.The bound(3)holds by Hölder's inequalities.Since a Banach space is a metric space with its norm, a continuous linear operator must be bounded. Conversely, any bounded linear operator must be continuous, because bounded operators preserve the Cauchy property of a Cauchy sequence.

### Reflexive space

Let be a normed space and denote the second dual vector space of . The canonical map defined by gives an isometric linear isomorphism (embedding) from into . The space is called reflexive if this map is surjective. This concept was introduced by Hahn (1927).For example, finite-dimensional (normed) spaces and Hilbert spaces are reflexive. The space of absolutely summable complex sequences is not reflexive. James (1951) constructed a non-reflexive Banach space that is isometrically isomorphic to its second conjugate space.Reflexive spaces are Banach spaces. This follows since given a normed space that may or may not be Banach, the norm on induces a norm (called the dual norm) on the dual of , and under the dual norm, is Banach. Iterating again, (the bidual of ) is also Banach, and since is reflexive if it coincides with its bidual, is Banach...

### Hilbert space

A Hilbert space is a vector space with an inner product such that the norm defined byturns into a complete metric space. If the metric defined by the norm is not complete, then is instead known as an inner product space.Examples of finite-dimensional Hilbert spaces include1. The real numbers with the vector dot product of and . 2. The complex numbers with the vector dot product of and the complex conjugate of . An example of an infinite-dimensional Hilbert space is , the set of all functions such that the integral of over the whole real line is finite. In this case, the inner product isA Hilbert space is always a Banach space, but theconverse need not hold.A (small) joke told in the hallways of MIT ran, "Do you know Hilbert? No? Then what are you doing in his space?" (S. A. Vaughn, pers. comm., Jul. 31, 2005)...

### Real normed algebra

A real normed algebra, also called a composition algebra, is a multiplication on that respects the length of vectors, i.e., for .The only real normed algebras with a multiplicative identity are the real numbers , complex numbers , quaternions , and octonions (Koecher and Remmert 1988).Hurwitz (1898) proved that a real normed algebra must have dimension , 2, 4, or 8. There are four real normed algebras of dimension 2: the complex numbers and three others (Koecher and Remmert 1988).Real normed algebras have no zero divisors since the equation implies that .

### Bounded approximation property

A Banach space has the approximation property (AP) if, for every and each compact subset of , there is a finite rank operator in such that for each , . If there is a constant such that for each such , , then is said to have bounded approximation property (BAP). For example, every Banach space with a Schauder basis has (BAP).

### Bicommutant theorem

The bicommutant theorem is a theorem within the field of functional analysis regarding certain topological properties of function algebras. The theorem says that, given a Hilbert space , a -subalgebra of which acts nondegenerately is dense in its bicommutant under the so-called sigma-strong operator topology. Here, denotes the algebra of bounded operators from to itself.The Bicommutant theorem is generally attributed to John von Neumann.The theorem itself has a number of important corollaries, not the least among which is an equivalence by which one can classify a subalgebra of as a von Neumann Algebra.

### Hilbert basis

A Hilbert basis for the vector space of square summable sequences , , ... is given by the standard basis , where , with the Kronecker delta. Thenwith . Although strictly speaking, the are not a vector basis because there exist elements which are not a finite linear combination, they are given the special term "Hilbert basis."In general, a Hilbert space has a Hilbert basis if the are an orthonormal basis and every element can be writtenfor some with .

### Bicommutant

Given a complex Hilbert space with associated space of continuous linear operators from to itself, the bicommutant of an arbitrary subset is the commutant of the commutant of , i.e., . In particular, the bicommutant is the collection of all elements in that commute with all elements of :Across the literature on the subject, the set is sometimes denoted , a reference to the fact that a linear operator between normed vector spaces is continuous if and only if it is bounded (Royden and Fitzpatrick 2010). Likewise, the bicommutant is sometimes called the double commutant.The notions of commutant and bicommutant are fundamental to the study of vonNeumann algebras (Dixmier 1981).

### Banach space

A Banach space is a complete vector space with a norm . Two norms and are called equivalent if they give the same topology, which is equivalent to the existence of constants and such that(1)and(2)hold for all .In the finite-dimensional case, all norms are equivalent. An infinite-dimensional space can have many different norms.A basic example is -dimensional Euclidean space with the Euclidean norm. Usually, the notion of Banach space is only used in the infinite dimensional setting, typically as a vector space of functions. For example, the set of continuous functions on closed interval of the real line with the norm of a function given by(3)is a Banach space, where denotes the supremum.On the other hand, the set of continuous functions on the unit interval with the norm of a function given by(4)is not a Banach space because it is not complete. For instance, the Cauchysequence of functions(5)does not converge to a continuous function.Hilbert..

### Prime banach space

A Banach space is called prime if each infinite-dimensional complemented subspace of is isomorphic to (Lindenstrauss and Tzafriri 1977).Pełczyński (1960) proved that (the Banach space of all complex sequences converging to zero together with the supremum norm) and for (the space of all complex sequences such that ) are prime. The L-infinity-space of all bounded complex sequences is also prime (Lindenstrauss 1967).

### Hardy space

If , then the Hardy space is the class of functions holomorphic on the disk and satisfying the growth conditionwhere is the Hardy norm.

### Banach limit

A Banach limit is a bounded linear functional on the space of complex bounded sequences that satisfies and for all .

### Banach completion

For a normed space , define to be the set of all equivalent classes of Cauchy sequences obtained by the relation(1)For and , let(2)(3)(4)Then is a Banach space containing a dense subspace that is isometric with . is called the (Banach) completion of (Kreyszig 1978).If is a normed algebra, makes into a Banach algebra. Moreover, if is a pre--algebra then equipped with is a -algebra (Murphy 1990).

### Grothendieck's theorem

Let and be paired spaces with a family of absolutely convex bounded sets of such that the sets of generate and, if , then there exists a such that and . Then is complete iff algebraic linear functional of that is weakly continuous on every is expressed as for some . When is not complete, the space of all linear functionals satisfying this condition gives the completion of .

### Banach algebra

A Banach algebra is an algebra over a field endowed with a norm such that is a Banach space under the norm and is frequently taken to be the complex numbers in order to ensure that the operator spectrum fully characterizes an operator (i.e., the spectral theorems for normal or compact normal operators do not, in general, hold in the operator spectrum over the real numbers).If is commutative and has a unit, then is invertible iff for all , where is the Gelfand transform.

### Positive linear functional

Let be a -algebra, then a linear functional on is said to be positive if it is a positive map, that is for all .Every positive linear functional is automatically continuous (Murphy 1990).

### Balanced set

A subset of a vector space is said to be balanced if whenever is a scalar satisfying . Here, the notation denotes the setSets which are balanced are sometimes called circled.

### Gelfand transform

The Gelfand transform is defined as follows. If is linear and multiplicative in the sensesandwhere is a commutative Banach algebra, then write . The Gelfand transform is automatically bounded.For example, if with the usual norm, then is a Banach algebra under convolution and the Gelfand transform is the Fourier transform. (In fact, may be replaced by any locally compact Abelian group, and then has a unit if and only if the group is discrete.)

### Baire function

The class of continuous functions is called the Baire class 0. For each , the functions that can be considered as pointwise limits of sequences of functions of Baire class but are not in any of the preceding classes are said to be of Baire class . A Baire or (analytically representable) function is that function belonging to a Baire class for some . For example, discontinuous functions representable by Fourier series belong to class 1.This notion was introduced by Baire in the 19th century. In 1905, Lebesgue showed that each of the Baire classes is nonempty and that there are (Lebesgue-) measurable functions that are not Baire functions (Kleiner 1989).

### Gelfand theorem

If is a locally compact T2-space, then the set of all continuous complex valued functions on vanishing at infinity (i.e., for each , the set is compact) equipped with the supremum norm is a commutative -algebra.The Gelfand theorem states that each commutative -algebra is of the form where is the maximal ideal space) of . is unital iff is compact.

### Poincar&eacute; inequality

Let be an open, bounded, and connected subset of for some and let denote -dimensional Lebesgue measure on . In functional analysis, the Poincaré inequality says that there exist constants and such thatfor all functions in the Sobolev space consisting of all functions in whose generalized derivatives are all also square integrable.This inequality plays an important role in the study of both function spaces and partial differential equations. As such, a number of generalizations have been established to domains and functions which are less well-behaved, e.g., to polyhedral domains and to functions which only have desirable behavior piecewise on .In some literature, the above-stated Poincaré inequality is sometimes referred to as the mean Poincaré inequality with the unqualified phrase "Poincaré inequality" reserved for the so-called (and closely-related) Friedrichs inequality. Inequalities..

### Approximation problem

The approximation problem is a well known problem of functional analysis (Grothendieck 1955). It asks to determine whether every compact operator from a Banach space to a Banach space is the uniform operator topology of a sequence of operators with finite rank. This question was answered in the negative by Enflo (1973), who provided a deep counterexample to this problem.