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Banach Spaces
In mathematics, Banach spaces 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.
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.
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 ƒ : [a, b] → K defined on a closed interval [a, b]
becomes a Banach space if we define the norm of such a function as ||ƒ|| = sup { |ƒ(x)| :
x ∈ [a, b] }, otherwise known as the supremum norm. This is indeed a well-defined
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 ƒ : [a, b] → K such that |ƒ|p
is Lebesgue integrable. The p-th root of this integral is then defined to be the norm of ƒ.
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: ƒ and g are equivalent if and
only if the norm of ƒ−g is zero. The set of equivalence classes then forms a Banach
space; it is denoted by Lp([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.
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 in
this way depends upon p.
If M is a closed linear 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.
If V and W are Banach spaces over the same ground field K, the set of all continuous Klinear 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.
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 infinitedimensional, 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.