In mathematics, Banach spaces (pronounced [ˈbanax], named after Stefan Banach) are one of the central objects of study in functional analysis. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and Stefan Banach ( Ukrainian: Степан Степанович Банах 1892–1945 was a Polish Mathematician who worked in interwar Poland and in For functional analysis as used in psychology see the Functional analysis (psychology article Many of the infinite-dimensional function spaces studied in analysis are examples of Banach spaces. In mathematics the dimension of a Space is roughly defined as the minimum number of Coordinates needed to specify every point within it In Mathematics, a function space is a set of functions of a given kind from a set X to a set Y.

## Definition

Banach spaces are defined as complete normed vector spaces. In Mathematical analysis, a Metric space M is said to be complete (or Cauchy) if every Cauchy sequence of points in M has In Mathematics, with 2- or 3-dimensional vectors with real -valued entries the idea of the "length" of a vector is intuitive and can easily be extended to This means that a Banach space is a vector space V over the real or complex numbers with a norm ||·|| such that every Cauchy sequence (with respect to the metric d(x, y) = ||xy||) in V has a limit in V. In Mathematics, a vector space (or linear space) is a collection of objects (called vectors) that informally speaking may be scaled and added In Mathematics, the real numbers may be described informally in several different ways Complex plane In Mathematics, the complex numbers are an extension of the Real numbers obtained by adjoining an Imaginary unit, denoted In Linear algebra, Functional analysis and related areas of Mathematics, a norm is a function that assigns a strictly positive length In Mathematics, a Cauchy sequence, named after Augustin Cauchy, is a Sequence whose elements become arbitrarily close to each other as the sequence In Mathematics, a metric or distance function is a function which defines a Distance between elements of a set. In Mathematics, the concept of a " limit " is used to describe the Behavior of a function as its argument either "gets close" Since the norm induces a topology on the vector space, a Banach space provides an example of a topological vector space. Topological spaces are mathematical structures that allow the formal definition of concepts such as Convergence, connectedness, and continuity. In Mathematics, a topological vector space is one of the basic structures investigated in Functional analysis.

## Examples

Throughout, let K stand for one of the fields R or C. In Abstract algebra, a field is an Algebraic structure in which the operations of Addition, Subtraction, Multiplication and division

The familiar Euclidean spaces Kn, where the Euclidean norm of x = (x1, . . . , xn) is given by ||x|| = (∑ |xi|2)1/2, are Banach spaces.

The space of all continuous functions f : [a, b] → K defined on a closed interval [a, b] becomes a Banach space if we define the norm of such a function as ||f|| = sup { |f(x)| : x in [a, b] }, otherwise known as the supremum norm. In Mathematics, a continuous function is a function for which intuitively small changes in the input result in small changes in the output In Mathematics, an interval is a set of Real numbers with the property that any number that lies between two numbers in the set is also included in the set In Mathematical analysis, the uniform norm assigns to real- or complex -valued bounded functions f the nonnegative number This is indeed a norm since continuous functions defined on a closed interval are bounded. The space is complete under this norm, and the resulting Banach space is denoted by C[a, b]. This example can be generalized to the space C(X) of all continuous functions XK, where X is a compact space, or to the space of all bounded continuous functions XK, where X is any topological space, or indeed to the space B(X) of all bounded functions XK, where X is any set. Topological spaces are mathematical structures that allow the formal definition of concepts such as Convergence, connectedness, and continuity. In all these examples, we can multiply functions and stay in the same space: all these examples are in fact unital Banach algebras. In Mathematics, an algebra is unital (some authors say unitary) if it contains a multiplicative Identity element (or unit) i In Mathematics, especially Functional analysis, a Banach algebra, named after Stefan Banach, is an Associative algebra A over the

For any open set Ω ⊆ C, the set A(Ω) of all bounded, analytic functions u : Ω → C is a complex Banach space with respect to the supremum norm. In Metric topology and related fields of Mathematics, a set U is called open if intuitively speaking starting from any point x in This article is about both real and complex analytic functions The fact that uniform limits of analytic functions are analytic is an easy consequence of Morera's theorem. In Complex analysis, a branch of Mathematics, Morera's theorem, named after Giacinto Morera, gives an important criterion for proving that a function is

If p ≥ 1 is a real number, we can consider the space of all infinite sequences (x1, x2, x3, . In Mathematics, a sequence is an ordered list of objects (or events . . ) of elements in K such that the infinite seriesi |xi|p is finite. In Mathematics, a series is often represented as the sum of a Sequence of terms That is a series is represented as a list of numbers with The p-th root of this series' value is then defined to be the p-norm of the sequence. The space, together with this norm, is a Banach space; it is denoted by l p.

The Banach space l consists of all bounded sequences of elements in K; the norm of such a sequence is defined to be the supremum of the absolute values of the sequence's members.

Again, if p ≥ 1 is a real number, we can consider all functions f : [a, b] → K such that |f|p is Lebesgue integrable. In Mathematics, the Integral of a non-negative function can be regarded in the simplest case as the Area between the graph of The p-th root of this integral is then defined to be the norm of f. By itself, this space is not a Banach space because there are non-zero functions whose norm is zero. We define an equivalence relation as follows: f and g are equivalent if and only if the norm of f - g is zero. In Mathematics, an equivalence relation is a Binary relation between two elements of a set which groups them together as being "equivalent" The set of equivalence classes then forms a Banach space; it is denoted by L p[a, b]. In Mathematics, given a set X and an Equivalence relation ~ on X, the equivalence class of an element a in X It is crucial to use the Lebesgue integral and not the Riemann integral here, because the Riemann integral would not yield a complete space. These examples can be generalized; see L p spaces for details. In Mathematics, the Lp and ℓp spaces are spaces of p-power integrable functions, and corresponding

If X and Y are two Banach spaces, then we can form their direct sum XY, which is again a Banach space. The symbol \oplus \! denotes direct sum it is also the astrological and astronomical symbol for Earth, and a symbol for the Exclusive disjunction This construction can be generalized to the direct sum of arbitrarily many Banach spaces.

If M is a closed subspace of the Banach space X, then the quotient space X/M is again a Banach space. The concept of a linear subspace (or vector subspace) is important in Linear algebra and related fields of Mathematics. In Linear algebra, the quotient of a Vector space V by a subspace N is a vector space obtained by "collapsing" N

Every inner product gives rise to an associated norm. In Mathematics, an inner product space is a Vector space with the additional Structure of inner product. The inner product space is called a Hilbert space if its associated norm is complete. This article assumes some familiarity with Analytic geometry and the concept of a limit. Thus every Hilbert space is a Banach space by definition. The converse statement also holds under certain conditions; see below.

## Linear operators

If V and W are Banach spaces over the same ground field K, the set of all continuous K-linear maps A : VW is denoted by L(V, W). In Topology and related areas of Mathematics a continuous function is a Morphism between Topological spaces Intuitively this is a function In Mathematics, a linear map (also called a linear transformation, or linear operator) is a function between two Vector spaces that Note that in infinite-dimensional spaces, not all linear maps are automatically continuous. L(V, W) is a vector space, and by defining the norm ||A|| = sup { ||Ax|| : x in V with ||x|| ≤ 1 } it can be turned into a Banach space.

The space L(V) = L(V, V) even forms a unital Banach algebra; the multiplication operation is given by the composition of linear maps. In Mathematics, especially Functional analysis, a Banach algebra, named after Stefan Banach, is an Associative algebra A over the

## Dual space

If V is a Banach space and K is the underlying field (either the real or the complex numbers), then K is itself a Banach space (using the absolute value as norm) and we can define the dual space V′ as V′ = L(V, K), the space of continuous linear maps into K. In Abstract algebra, a field is an Algebraic structure in which the operations of Addition, Subtraction, Multiplication and division In Mathematics, the real numbers may be described informally in several different ways Complex plane In Mathematics, the complex numbers are an extension of the Real numbers obtained by adjoining an Imaginary unit, denoted In Mathematics, the absolute value (or modulus) of a Real number is its numerical value without regard to its sign. In Mathematics, any Vector space V has a corresponding dual vector space (or just dual space for short consisting of all Linear functionals This is again a Banach space (with the operator norm). In Mathematics, the operator norm is a means to measure the "size" of certain Linear operators Formally it is a norm defined on the space of It can be used to define a new topology on V: the weak topology. Topology ( Greek topos, "place" and logos, "study" is the branch of Mathematics that studies the properties of In Mathematics, weak topology is an alternative term for Initial topology.

Note that the requirement that the maps be continuous is essential; if V is infinite-dimensional, there exist linear maps which are not continuous, and therefore not bounded, so the space V* of linear maps into K is not a Banach space. In Mathematics, a function f defined on some set X with real or complex values is called bounded, if the set The space V* (which may be called the algebraic dual space to distinguish it from V') also induces a weak topology which is finer than that induced by the continuous dual since V′⊆V*. In Topology and related areas of Mathematics comparison of topologies refers to the fact that two Topological structures on a given set may stand in relation

There is a natural map F from V to V′′ (the dual of the dual) defined by

F(x)(f) = f(x)

for all x in V and f in V′. Because F(x) is a map from V′ to K, it is an element of V′′. The map F: xF(x) is thus a map VV′′. As a consequence of the Hahn-Banach theorem, this map is injective; if it is also surjective, then the Banach space V is called reflexive. In Mathematics, the Hahn–Banach theorem is a central tool in Functional analysis. In Mathematics, a function f is said to be surjective or onto, if its values span its whole Codomain; that is for every In Functional analysis, a Banach space is called reflexive if it satisfies a certain abstract property involving Dual spaces Reflexive spaces turn out to Reflexive spaces have many important geometric properties. A space is reflexive if and only if its dual is reflexive, which is the case if and only if its unit ball is compact in the weak topology. In Mathematics, weak topology is an alternative term for Initial topology.

For example, lp is reflexive for 1<p<∞ but l1 and l are not reflexive. The dual of lp is lq where p and q are related by the formula (1/p) + (1/q) = 1. See L p spaces for details. In Mathematics, the Lp and ℓp spaces are spaces of p-power integrable functions, and corresponding

## Relationship to Hilbert spaces

As mentioned above, every Hilbert space is a Banach space because, by definition, a Hilbert space is complete with respect to the norm associated with its inner product, where a norm and an inner product are said to be associated if ||v||² = (v,v) for all v. This article assumes some familiarity with Analytic geometry and the concept of a limit.

The converse is not always true; not every Banach space is a Hilbert space. A necessary and sufficient condition for a Banach space V to be associated to an inner product (which will then necessarily make V into a Hilbert space) is the parallelogram identity:

$\|u+v\|^2 + \|u-v\|^2 = 2(\|u\|^2 + \|v\|^2)$

for all u and v in V, and where ||*|| is the norm on V. In Mathematics, the simplest form of the parallelogram law belongs to elementary Geometry. So, for example, while Rn is a Banach space with respect to any norm defined on it, it is only a Hilbert space with respect to the Euclidean norm. Similarly, as an infinite-dimensional example, the Lebesgue space Lp is always a Banach space but is only a Hilbert space when p = 2.

If the norm of a Banach space satisfies this identity, the associated inner product which makes it into a Hilbert space is given by the polarization identity. If V is a real Banach space, then the polarization identity is

$\langle u,v\rangle = \frac{1}{4} (\|u+v\|^2 - \|u-v\|^2)$

whereas if V is a complex Banach space, then the polarization identity is given by

$\langle u,v\rangle = \frac{1}{4} \left(\|u+v\|^2 - \|u-v\|^2 + i(\|u+iv\|^2 - \|u-iv\|^2)\right).$

The necessity of this condition follows easily from the properties of an inner product. To see that it is sufficient—that the parallelogram law implies that the form defined by the polarization identity is indeed a complete inner product—one verifies algebraically that this form is additive, whence it follows by induction that the form is linear over the integers and rationals. Mathematical induction is a method of Mathematical proof typically used to establish that a given statement is true of all Natural numbers It is done by proving that Then since every real is the limit of some Cauchy sequence of rationals, the completeness of the norm extends the linearity to the whole real line. In the complex case, one can check also that the bilinear form is linear over i in one argument, and conjugate linear in the other.

## Hamel dimension

It follows from the completeness of Banach spaces and the Baire category theorem that a Hamel basis of an infinite-dimensional Banach space is uncountable. The Baire category theorem is an important tool in General topology and Functional analysis. Basis vector redirects here For basis vector in the context of crystals see Crystal structure.

## Derivatives

Several concepts of a derivative may be defined on a Banach space. See the articles on the Fréchet derivative and the Gâteaux derivative. In Mathematics, the Fréchet derivative is a Derivative defined on Banach spaces Named after Maurice Fréchet, it is commonly used to formalize In Mathematics, the Gâteaux differential is a generalisation of the concept of Directional derivative in Differential calculus.

## Generalizations

Several important spaces in functional analysis, for instance the space of all infinitely often differentiable functions RR or the space of all distributions on R, are complete but are not normed vector spaces and hence not Banach spaces. In Mathematical analysis, distributions (also known as generalized functions) are objects which generalize functions and Probability distributions In Fréchet spaces one still has a complete metric, while LF-spaces are complete uniform vector spaces arising as limits of Fréchet spaces. This article deals with Fréchet spaces in functional analysis In Mathematics, a metric space is a set where a notion of Distance (called a metric) between elements of the set is defined In Mathematics, an LF -space is a Topological vector space V that is a countable strict Inductive limit of Fréchet In the Mathematical field of Topology, a uniform space is a set with a uniform structure.

## Literature

Historical monographs in English, French and Polish: