Banach space
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In mathematics, Banach spaces (pronounced [ˈbanax], named after Polish mathematician Stefan Banach) are one of the central objects of study in functional analysis. They are topological vector spaces that have many interesting properties associated with them. For instance, one can model topological spaces on Banach spaces (just as one models topological spaces on Euclidean space). Such spaces are known as banach manifolds. Many of the infinite-dimensional function spaces studied in analysis are examples of Banach spaces.
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[edit] 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, every Banach space is necessarily metrizable and metrizable spaces generally have very interesting properties.
[edit] Examples
Throughout, let K stand for one of the fields R or C.
The familiar Euclidean spaces Kn, where the Euclidean norm of x = (x1, ..., xn) is given by ||x|| = (∑i=1...n |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. 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.
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.
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 |xi|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.
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.
If X and Y are two Banach spaces, then we can form their direct sum X ⊕ Y, which has a natural topological vector space structure but no canonical norm. However, it is again a Banach space for several equivalent norms, for example
This construction can be generalized to define ℓp-direct sums of arbitrarily many Banach spaces. When there is an infinite number of non-zero summands, the space obtained depends upon p.
If M is a closed subspace of the Banach space X, then the quotient space X/M is again a Banach space.
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.
[edit] 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.
The space L(V) = L(V, V) even forms a unital Banach algebra; the multiplication operation is given by the composition of linear maps.
[edit] 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.
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∗.
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, and isometric; 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.
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.
[edit] 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:
for all u and v in V, and where ||*|| is the norm on V. 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
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.
[edit] 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.
[edit] Derivatives
Several concepts of a derivative may be defined on a Banach space. See the articles on the Fréchet derivative and the Gâteaux derivative.
[edit] 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.
[edit] See also
[edit] Literature
Historical monographs in English, French and Polish:
- Stefan Banach: Théorie des opérations linéaires. -- Warszawa 1932. (Monografie Matematyczne; 1) Zbl 0005.20901
