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Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vectors and is complete in the sense that a Cauchy sequence of vectors always converges to a well defined limit that bwnach within the space.
Banach spaces are named after the Polish mathematician Stefan Banachwho introduced this concept and studied it systematically in — along with Hans Hahn and Eduard Helly. Banach spaces play a central role in functional analysis. In other areas of analysisthe spaces under study are often Banach spaces.
The vector space structure allows one to relate the behavior of Cauchy sequences to that of converging series of vectors. A normed space X is a Banach space if and only if each absolutely convergent series in X converges, .
Completeness of a normed space is preserved if the given norm is replaced by eslacio equivalent one. All norms on a finite-dimensional vector espaacio are equivalent.
Every finite-dimensional normed space over R or C is a Banach space. If X and Y are normed spaces over the same ground field Kthe set of all continuous K -linear maps Banacg In infinite-dimensional spaces, not banacg linear maps are continuous.
A linear mapping from a normed space X to another normed space is continuous if and only if it is bounded banac the closed unit ball of X. Thus, the vector space B XY can be given the operator norm. If X and Y are normed spaces, they are isomorphic banafh spaces if there exists a linear bijection T: If one of the two spaces X or Y is complete or reflexiveseparableetc.
Two normed spaces X and Y are isometrically isomorphic if in addition, T is an isometryi. Every normed space X can be isometrically embedded in a Banach space. More precisely, for every normed space Xthere exist a Banach space Y and a mapping T: If Z is another Banach space such that there is an isometric isomorphism from X onto a dense subset of Zthen Z is isometrically isomorphic to Y.
This Banach space Y is the completion of the normed space X. The underlying metric space for Y is the same as the metric completion of Xwith the vector space operations extended from X to Y. However, several equivalent norms espacip commonly used,  such as.
Normed vector space
The closed linear subspace M of X is said to be a complemented subspace of X if M is the range of a bounded linear projection P from X onto M. In this case, the space X is isomorphic to the direct sum of M and Ker Pthe kernel of the projection P. There wspacio a canonical factorization of T as . This is well defined because all elements in the same class have the same image.
Basic examples  of Banach spaces include: According to the Banach—Mazur theoremevery Banach space is isometrically isomorphic to a subspace of some C K. Any Hilbert space serves as an example of a Banach space. The Hardy spacesthe Sobolev spaces are examples of Banach spaces that are related to L espwcio spaces and have additional structure.
They are important in different branches of analysis, Harmonic analysis and Partial differential equations among others. If X is a normed space and K the underlying field either the real or the complex numbersthe continuous dual space is the space of continuous linear maps from X into Kor continuous linear functionals. The main tool for proving the existence of continuous linear functionals is the Hahn—Banach theorem.
Cauchy sequence – Wikipedia
In particular, every continuous linear functional on a subspace of a normed space can be continuously extended to the whole space, without increasing the norm espaico the functional. The Hahn—Banach separation theorem states that two disjoint non-empty convex sets in a real Banach space, one of them open, can be separated by a closed affine hyperplane.
The open convex set lies strictly on one side of the hyperplane, the second convex set lies espacoi the other side but may touch the hyperplane. The subset S is total in X if and only if the only continuous linear functional that vanishes on S is the 0 functional: The norm topology is therefore finer than the weak topology. It follows from the Hahn—Banach separation theorem that the weak topology is Hausdorffand that a norm-closed convex bancah of a Banach space is also weakly closed.
If X is infinite-dimensional, there exist linear maps which are not continuous. Its importance comes from the Banach—Alaoglu theorem. The Banach—Alaoglu theorem depends on Tychonoff’s theorem about infinite products of compact spaces.
For every vector y in a Hilbert espcaio Hthe mapping.
When the scalars are real, this map is an isometric isomorphism. The extreme points of P K are the Dirac measures on K. In the commutative Banach algebra C Kthe maximal ideals are precisely kernels of Dirac mesures on K.
More generally, by the Gelfand—Mazur theoremthe maximal ideals of a unital commutative Banach algebra can be identified with its characters —not merely as sets but as topological spaces: For every normed space Xthere is a natural map. The map F X: It is indeed isometric, but not onto. If F X is surjectivethen the normed space X is called reflexive see below. Using the isometric embedding F Xit is customary to consider a normed space X as a subset of its bidual.
Here are the main general results about Banach spaces that go back to the time of Banach’s book Banach and are related to the Baire category theorem. Therefore, a Banach space cannot be the union of countably many closed subspaces, unless it is already equal to one of them; a Banach space with a countable Hamel basis is finite-dimensional. The Banach—Steinhaus theorem is not limited to Banach spaces. This result is a direct consequence of the preceding Banach isomorphism theorem and of the canonical factorization of bounded linear maps.
The normed space X is called reflexive when the natural map.
This is a consequence of the Hahn—Banach theorem. Further, by the open mapping theorem, if there is a bounded linear operator from the Banach space X onto the Banach space Ythen Y is reflexive. Hilbert spaces are reflexive. More generally, uniformly convex spaces are reflexive, by the Milman—Pettis theorem. Furthermore, this space J is isometrically isomorphic to its bidual.
When X is reflexive, it follows that all closed and bounded convex subsets of X are weakly compact. In a Hilbert space Hthe weak compactness of the unit ball is very often used in the following way: Weak compactness of the unit ball provides a tool for finding solutions in reflexive spaces to certain optimization problems.
For example, every convex continuous function on the unit ball B of a reflexive space attains its minimum at some point in B. The following theorem of Robert C. James provides a converse statement. On every non-reflexive Banach space Xthere exist continuous linear functionals that are not norm-attaining.
The Banach space X is weakly sequentially complete if every weakly Cauchy sequence is weakly convergent in X. It follows from the preceding discussion that reflexive spaces are weakly sequentially complete. An orthonormal sequence in a Hilbert space is a simple example of a weakly convergent sequence, with limit equal to the 0 vector. For every weakly null sequence in a Banach space, there exists a sequence of convex combinations of vectors from the given sequence that is norm-converging to 0.
The unit ball of the bidual is a pointwise compact subset of the first Baire class on K. This applies to separable reflexive spaces, but more is true in this case, as stated below.
The weak topology of a Banach space X is metrizable if and only if X is finite-dimensional. This applies in particular to separable reflexive Banach spaces.
Although the weak topology of the unit ball is not metrizable in general, one can characterize weak compactness using sequences. A Banach space X is reflexive if and only if each bounded sequence in X has a weakly convergent subsequence. Banach spaces with a Schauder basis are necessarily separablebecause the countable set of finite linear combinations with rational coefficients say is dense.
They are called biorthogonal functionals. Most classical separable spaces have explicit bases. The Schauder system is a basis in the space C [0, 1]. Since every vector x in a Banach space X with a basis is the limit of P n xwith P n of finite rank and uniformly bounded, the space X satisfies the bounded approximation property. The first example by Enflo of a space failing the approximation property was at the same time the first example of a separable Banach space without a Schauder basis.
James characterized reflexivity epsacio Banach spaces with a basis: Let X and Y be two K -vector spaces. There are various norms that can be placed on the tensor product of the underlying vector spaces, espscio others the projective cross norm and injective cross norm introduced by A.
In general, the tensor product of complete spaces is not complete again. Grothendieck proved in particular that . Let X be a Banach space. When X has the approximation propertythis closure coincides with the space of compact operators on X.
For every Banach space Ythere is a natural norm 1 linear map. Grothendieck related the approximation problem to the question of whether this map is one-to-one when Y is the dual of X. Precisely, for every Banach space Xthe map. This was disproved by Gilles Pisier in Furthermore, just as Enflo’s example, this space X is a “hand-made” space that fails to have the approximation property. A necessary and sufficient condition for the norm of a Banach space X to be associated to an inner product is the parallelogram identity:.
If this identity is satisfied, the associated inner product is given by the polarization identity.