Banach Space

Know this topic well? Help others and get FREE products!

Banach Space

A Banach space is a vector space over the field of real numbers, or over the field of complex numbers, together with a norm. And in a Banach space the metric topology determine by the norm is complete. Thus the assertion that a certain object is a Banach space includes several different pieces of information.

We now describe a Banach space more carefully. We begin with a vector space X having either the real numbers or the complex numbers as its field of scalars, (see the article on abstract linear spaces.) By a norm on X we understand a function ∥ ∥: X [0, ) that satisfies the following three conditions:

• (1) ∥x∥ = 0 if and only if x = 0 is the zero vector in X,
• (2) ∥x∥ = || ∥x∥ for all scalars and all vectors x,
• (3) ∥x + y∥ ∥x∥ + ∥y∥ for all pairs of vectors x and y.

The norm ∥ ∥ on X allows us to define a closely related metric on X. We say that the distance from x to y is given by ∥x - y∥. Alternatively, the function from X x X into [0, ) defined by sending (x, y) ∥x - y∥ is a metric (see the article on metric spaces.) This metric automatically induces a metric topology in the vector space X. If y is a point in X and r > 0 then we recall that the open ball centered at y and having radius r is the set

B_r(y) = {x ∈ X: ∥x - y∥ < r}.

More generally, a set in the metric topology is open if it is empty, or if it is a union of open balls. The pair (X, ∥ ∥) is often called a normed linear space. It consists of the vector space X over the field of real numbers or complex numbers, the norm function ∥ ∥, and the metric topology in X which is induced by the norm. Thus the set X is both a vector space and a metric space.

Next we recall that a sequence {x_n}, where n = 1, 2, 3, ... , in the metric space X is called a Cauchy sequence if for every ε > 0 there exists a positive integer N such that

∥x_n - x_m∥ < ε whenever N m and N n.

If the sequence {x_n} is convergent in X then it is easy to prove that it must be a Cauchy sequence. However, in general, a metric space may have Cauchy sequences which do not converge to a point in the space. We say that a metric space, or the metric itself, is complete if every Cauchy sequence does converge to a point in the space. We are now in position to give the definition of a Banach space: a Banach space is a normed linear space over the field of real numbers or over the field of complex number, such the metric determined by the norm is complete. Therefore, in a Banach space, every Cauchy sequence must converge to a point in the space.

Important examples of Banach spaces are the finite dimensional Banach spaces. For example, we could use the vector space ℜ^N of all column vectors

in which each coordinate x_n, 1 n N, is an element from the field ℜ of real numbers. Next we need to specify a norm on ℜ^N. This can be done in many ways, for example we could define the norm ∥ ∥₁ by setting

∥x∥₁ = |x₁| + |x₂| + ... + |x_N|.

Then it is easy to check that ∥ ∥₁ satisfies the three conditions of a norm, and the resulting metric topology in ℜ^N is in fact complete. Thus the pair (ℜ^N, ∥ ∥₁) forms a Banach space. Another example of a norm on ℜ^N is the Euclidean norm, which is often denoted by ∥ ∥₂ and then defined by

∥x∥₂ = (|x₁|² + |x₂|² + ... + |x_N|²)^½.

More generally, if p is a real number and 1 p < then each of the functions ∥ ∥_p: ℜ^N [0, ) defined by

is a norm on ℜ^N. There is also a norm denoted by ∥ ∥ and then defined by

∥x∥ = max{|x₁|, |x₂|, ... , |x_N|}.

It can be shown that for each p with 1 p , the metric topology determined by ∥ ∥_p is complete. That is, for each p with 1 p the pair (ℜ^N, ∥ ∥_p) forms a Banach space.

Here is another example of a Banach space. Let C[0,1] denote the set of all continuous, complex valued functions on the closed interval [0,1]. If and are complex numbers, if ƒ and g are functions in the set C[0,1], then the linear combination ƒ(x) + g(x) turns out to be another function in C[0,1]. From this observation it is easy to see that C[0,1] forms a vector space over the field of complex numbers. Now define ∥ ∥:C[0,1] [0, ) by

∥ƒ∥ = sup{|ƒ(x)|: 0 x 1}.

It turns out that ∥ ∥ defines a norm on C[0,1]. And by using the fact that a continuous function on the compact interval [0,1] is uniformly continuous, it can be shown that every sequence in C[0,1] which is Cauchy with respect to the norm ∥ ∥, converges to a function in C[0,1]. Thus the vector space C[0,1] together with the norm ∥ ∥ forms a Banach space.

Another method of constructing a norm on the vector space C[0,1] is to define

Here we can use the Riemann integral from calculus, since the functions ƒ in C[0,1] are continuous and therefore Riemann integrable on the interval [0,1]. While ∥ ∥₁ does satisfy the requirements of a norm on C[0,1], there exist sequences of functions in C[0,1] which are Cauchy with respect to this norm but which do not converge to a function in C[0,1]. Thus the pair (C[0,1], ∥ ∥) forms a normed linear space but it is not a Banach space. This example raises a natural question: is it possible to enlarge the vector space C[0,1], and extend the definition of the norm ∥ ∥₁ to the enlarged space, in such a way that we do get a Banach space? The answer to this question is yes. The enlarged vector space turns out to be the collection of equivalence classes of Lebesgue integrable functions on the interval [0,1]. The extended norm is defined in the same manner but now the Lebesgue integral must be used because, in general, the absolute value of the functions are no longer Riemann integrable.

This is the complete article, containing 1,075 words (approx. 4 pages at 300 words per page).