Banach Space Definition Examples Linear Operators

In mathematics, Banach spaces, named after Stefan Banach who studied them, are one of the central objects of study in functional analysis. Banach spaces are typically infinite-dimensional spaces containing functions.

1 Definition

2 Examples

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.

The space of all continuous functions f : [a, b] → K defined on a closed intervalTopology In elementary algebra, an interval is a set that contains every real number between two indicated numbers, and possibly the two numbers themselves. Interval notation is where the permitted values for a variable are expressed as ranging over an in [a, b] becomes a Banach space if we define the norm of such a function as

||f|| = sup { |f(x)| : x in [a, b] }. 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 spaceIn mathematics, a compact space is a space that resembles a closed and bounded subset of Euclidean space R n in that it is "small" in a certain sense and "contains all its limit points". The modern general definition calls a topological space compact if e, or to the space of all bounded continuous functions X → K, where X is any topological spaceTopological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion. The branch of mathematics that studies, or indeed to the space B(X) of all bounded functions X → K, where X is any setThis article is about sets in mathematics. For other meanings, see Set (disambiguation). Sets are one of the most important and fundamental concepts in modern mathematics. Basic set theory, having only been invented at the end of the 19th century, is now. In all these examples, we can multiply functions and stay in the same space: all these examples are in fact unitary Banach algebrasIn 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. The algebra multiplication and the Banach space norm are required to be related.

If p ≥ 1 is a real number, we can consider the space of all infinite sequenceThis is a page about mathematics. For other usages of "sequence", see: sequence (non-mathematical). In mathematics, a sequence is a list of objects (or events) which have been ordered in a sequential fashion; such that each member either comes before, ors (x₁, x₂, x₃, ...) of elements in K such that the infinite series ∑ |x_i|^p converges. 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 is again a Banach space. 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.

3 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 unitary Banach algebra; the multiplication operation is given by the composition of linear maps.

4 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 by V = L(V, K). This is again a Banach space. It can be used to define a new topology on V: the weak topology.

There is a natural map F from V to V defined by

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

for all x in V and f in 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.

For example, l^p is reflexive for 1 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.
5 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 to its inner product.
A necessary and sufficient condition for the converse to be true, i.e. for a Banach space to also be a Hilbert space is the parallelogram identity:

for all u and v in our Banach space V, and where ||*|| is the norm on V.
If the norm of a Banach space satisfies this identity, then the space is also a Hilbert space, with inner product 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

to see why the parallelogram 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.

6 Derivatives

It is possible to define the derivative of a function f : V → W between two Banach spaces. Intuitively, if x is an element of V, the derivative of f at the point x is a continuous linear map which approximates f near x.
Formally, f is called differentiable at x if there exists a continuous linear map A : V → W such that

The limit here is taken over all sequences of non-zero elements in V which converge to 0. If the limit exists, we write Df(x) = A and call it the derivative of f at x.
This notion of derivative is in fact a generalization of the ordinary derivative of functions R → R, since the linear maps from R to R are just multiplications with real numbers.
If f is differentiable at every point x of V, then Df : V → L(V, W) is another map between Banach spaces (in general not a linear map!), and can possibly be differentiated again, thus defining the higher derivatives of f. The n-th derivative at a point x can then be viewed as a multilinear map Vⁿ → W.
Differentiation is a linear operation in the following sense: if f and g are two maps V → W which are differentiable at x, and r and s are scalars from K, then rf + sg is differentiable at x with D(rf + sg)(x) = rD(f)(x) + sD(g)(x).
The chain rule is also valid in this context: if f : V → W is differentiable at x in V, and g : W → X is differentiable in f(x), then the composition g o f is differentiable in x and the derivative is the composition of the derivatives:

7 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.

8 External links

For historical references see the Banach space entry in

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

9 See also

wikibooks:Algebra:Inner product spaces

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Home > Banach space

1 Definition

2 Examples

3 Linear operators

4 Dual space

5 Relationship to Hilbert spaces

6 Derivatives

7 Generalizations

8 External links

9 See also

Topics: Banach Space, Definition, Examples, Linear Operators...