In mathematics, Banach spaces (pronounced ['banaɣ]), named after Stefan Banach who studied them, are one of the central objects of study in functional analysis. Many of the infinite-dimensional function spaces studied in functional analysis are examples of Banach spaces. Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ... Stefan Banach Stefan Banach (March 30, 1892 in Kraków, Austria-Hungary now Poland– August 31, 1945 in Lwów, Soviet Union - occupied Poland), was an eminent Polish mathematician, one of the moving spirits of the Lwów School of Mathematics in pre-war Poland. ... Functional analysis is the branch of mathematics, and specifically of analysis, concerned with the study of spaces of functions. ... For other uses, see Dimension (disambiguation). ... In mathematics, a function space is a set of functions of a given kind from a set X to a set Y. It is called a space because in many applications, it is a topological space or a vector space or both. ...

Definition

Banach spaces are defined as complete normed vector spaces. 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) = ||x − y||) in V has a limit in V. Since the norm induces a topology on the vector space, a Banach space provides an example of a topological vector space. In mathematical analysis, a metric space M is said to be complete (or Cauchy) if every Cauchy sequence of points in M has a limit that is also in M. Intuitively, a space is complete if it doesnt have any holes, if there arent any points missing. For... 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 any real vector space Rn. ... 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 as numbers that can be given by an infinite decimal representation, such as 2. ... In mathematics, a complex number is a number of the form where a and b are real numbers, and i is the imaginary unit, with the property i 2 = −1. ... 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 any real vector space Rn. ... In mathematical analysis, a Cauchy sequence, named after Augustin Cauchy, is a sequence whose elements become close as the sequence progresses. ... In mathematics, a metric space is a set where a notion of distance between elements of the set is defined. ... Wikibooks Calculus has a page on the topic of Limits In mathematics, the concept of a limit is used to describe the behavior of a function as its argument either gets close to some point, or as it becomes arbitrarily large; or the behavior of a sequences elements, as... Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ... In mathematics a topological vector space is one of the basic structures investigated in functional analysis. ...

It follows from the completeness requirement and Baire category theorem that the cardinality of a Hamel basis for an (infinite dimensional) Banach space is uncountable. The Baire category theorem is an important tool in general topology and functional analysis. ... In mathematics, a subset B of a vector space V is said to be a basis of V if it satisfies one of the four equivalent conditions: B is both a set of linearly independent vectors and a generating set of V. B is a minimal generating set of V...

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 (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ...

The familiar Euclidean spaces Kⁿ, where the Euclidean norm of x = (x₁, ..., x_n) is given by ||x|| = (∑ |x_i|²)^1/2, are Banach spaces. Around 300 BC, the Greek mathematician Euclid laid down the rules of what has now come to be called Euclidean geometry, which is the study of the relationships between angles and distances in space. ...

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. 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 X → K, where X is a compact space, or to the space of all bounded continuous functions X → K, where X is any topological space, or indeed to the space B(X) of all bounded functions X → K, where X is any set. 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, a continuous function is a function for which, intuitively, small changes in the input result in small changes in the output. ... In mathematics, interval is a concept relating to the sequence and set-membership of one or more numbers. ... In mathematical analysis, the uniform norm assigns to real- or complex-valued functions f the nonnegative number This norm is also called the supremum norm or the Chebyshev norm. ... In mathematics, a subset of Euclidean space Rn is called compact if it is closed and bounded. ... Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ... In mathematics, a set can be thought of as any collection of distinct objects considered as a whole. ... In mathematics, an associative algebra is unital if it contains a multiplicative identity element, i. ... In functional analysis, a Banach algebra, named after Stefan Banach, is an associative algebra A over the real or complex numbers which at the same time is also a Banach space. ...

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. The fact that uniform limits of analytic functions are analytic is an easy consequence of Morera's theorem. In topology and related fields of mathematics, a set U is called open if, intuitively speaking, you can wiggle or change any point x in U by a small amount in any direction and still be inside U. In other words, if x is surrounded only by elements of U... In mathematics, an analytic function is a function that is locally given by a convergent power series. ... In complex analysis, Moreras theorem states that if the integral of a continuous complex-valued function f of a complex variable along every simple closed curve within an open set is zero, that is, if for C any simple closed curve, then f is differentiable at every point in...

If p ≥ 1 is a real number, we can consider the space of all infinite sequences (x₁, x₂, x₃, ...) of elements in K such that the infinite series ∑_i |x_i|^p is finite. 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. For other senses of this word, see sequence (disambiguation). ... In mathematics, a series is a sum of a sequence of terms. ...

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. 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. The set of equivalence classes then forms a Banach space; it is denoted by L ^p[a, b]. 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. The integral of a positive function can be interpreted as the area under a curve. ... In mathematics, an equivalence relation is a binary relation between two elements of a set which groups them together as being equivalent in some way. ... In mathematics, given a set X and an equivalence relation ~ on X, the equivalence class of an element a in X is the subset of all elements in X which are equivalent to a: [a] = { x ∈ X | x ~ a } The notion of equivalence classes is useful for constructing sets out... In mathematics, the Lp and spaces are spaces of p-power integrable functions, and corresponding sequence spaces. ...

If X and Y are two Banach spaces, then we can form their direct sum X ⊕ Y, which is again a Banach space. This construction can be generalized to the direct sum of arbitrarily many Banach spaces. In abstract algebra, the direct sum is a construction which combines several vector spaces (or groups, or abelian groups, or modules) into a new, bigger one. ...

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 to zero. ...

Every inner product gives rise to an associated norm. The inner product space is called a Hilbert space if its associated norm is complete. Thus every Hilbert space is a Banach space by definition. The converse statement also holds under certain conditions; see below. In mathematics, an inner product space is a vector space with additional structure, an inner product (also called a scalar product), which allows us to introduce geometrical notions such as angles and lengths of vectors. ... The mathematical concept of a Hilbert space (named after the German mathematician David Hilbert) generalizes the notion of Euclidean space in a way that extends methods of vector algebra from the plane and three-dimensional space to spaces of functions. ...

Linear operators

If V and W are Banach spaces over the same ground field K, the set of all continuous K-linear maps A : V → W is denoted by L(V, W). 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. In topology and related areas of mathematics a continuous function is a morphism between topological spaces. ... In mathematics, a linear transformation (also called linear map or linear operator) is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication. ...

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 functional analysis, a Banach algebra, named after Stefan Banach, is an associative algebra A over the real or complex numbers which at the same time is also a Banach space. ...

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. This is again a Banach space (with the operator norm). It can be used to define a new topology on V: the weak topology. In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ... In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ... In mathematics, a complex number is a number of the form where a and b are real numbers, and i is the imaginary unit, with the property i 2 = −1. ... In mathematics, the absolute value (or modulus[1]) of a real number is its numerical value without regard to its sign. ... In mathematics, the existence of a dual vector space reflects in an abstract way the relationship between row vectors (1×n) and column vectors (n×1). ... In mathematics, the operator norm is a means to measure the size of certain linear operators. ... A Möbius strip, an object with only one surface and one edge; such shapes are an object of study in topology. ... 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. 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 mathematics, a function f defined on some set X with real or complex values is called bounded, if the set of its values is bounded. ... In mathematics, the possible topologies on a given set X form a partially ordered set: if a collection τ1 of subsets of X contains each subset in a collection τ2, and these are both topologies on X, we say that τ1 is a finer (alt. ...

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: x → F(x) is thus a map V → V′′. 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. 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, the Hahn-Banach theorem is a central tool in functional analysis. ... In mathematics, an injective function (or one-to-one function or injection) is a function which maps distinct input values to distinct output values. ... In mathematics, a surjective function (or onto function or surjection) is a function with the property that all possible output values of the function are generated when the input ranges over all the values in the domain. ... This page concerns the reflexivity of a Banach space. ... In mathematics, a subset of Euclidean space Rn is called compact if it is closed and bounded. ... In mathematics, weak topology is an alternative term for initial topology. ...

For example, l^p is reflexive for 1<p<∞ but l¹ and l^∞ are not reflexive. The dual of l^p is l^q 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 spaces are spaces of p-power integrable functions, and corresponding sequence spaces. ...

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.

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: The parallelogram law in elementary geometry In elementary geometry, the parallelogram law states that the sum of the squares of the lengths of the fours sides of a parallelogram equals the sum of the squares of the lengths of the two diagonals. ...

for all u and v in V, and where ||*|| is the norm on V.

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

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

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. 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. Mathematical induction is a method of mathematical proof typically used to establish that a given statement is true of all natural numbers. ...

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. ... In mathematics, the Gâteaux derivative is a generalisation in functional analysis of the standard derivative of the differential calculus. ...

Generalizations

Several important spaces in functional analysis, for instance the space of all infinitely often differentiable functions R → R or the space of all distributions on R, are complete but are not normed vector spaces and hence not Banach spaces. In Fréchet spaces one still has a complete metric, while LF-spaces are complete uniform vector spaces arising as limits of Fréchet spaces. In mathematical analysis, distributions (also known as generalized functions) are objects which generalize functions and probability distributions. ... This article deals with Fréchet spaces in functional analysis. ... In mathematics, a metric space is a set where a notion of distance 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 spaces. ... In the mathematical field of topology, a uniform space is a set with a uniform structure. ...

Literature

Historical monographs in English, French and Polish:

• Stefan Banach: Théorie des opérations linéaires. -- Warszawa 1932. (Monografie Matematyczne; 1) Zbl 0005.20901

Stefan Banach Stefan Banach (March 30, 1892 in Kraków, Austria-Hungary now Poland– August 31, 1945 in Lwów, Soviet Union - occupied Poland), was an eminent Polish mathematician, one of the moving spirits of the Lwów School of Mathematics in pre-war Poland. ...

External links

For historical references see the Banach space entry in

• Earliest known uses of some of the words of mathematics: B