Question

Show that cvery space \(F(S)\) is complete with respect to the supremum norm of Example 10.26. Hence show that the vector space \(\ell_{\infty}\) of bounded infinite complex sequences is a Banach space with respect to the norm \(\|\mathrm{x}\|=\sup \left(x_{t}\right)\).

Short answer

By showing that the function space \(F(S)\) is complete with respect to supremum norm, one can infer the completeness of the space of infinite complex sequences \(\ell_{\infty}\) under the same norms, hence, proving it to be a Banach space.

Step-by-step solution

Understand the supremum norm

The supremum norm is a particular type of the infinity norm. For a given function \(f\), the supremum norm is designated as \(\|f\|\_\infty\) and defined as the supremum (the least upper bound) of the absolute values of its elements, i.e., \(\|f\|\_\infty = \sup\|f(x)\|\). In function space \(F(S)\), it would be \(\|f\|\_\infty = \sup_{x \in S}\|f(x)\|\).

Prove that \(F(S)\) is complete with respect to supremum norm

Assume \((f_n)\) is a Cauchy sequence in \(F(S)\) under the supremum norm. This means that for every \(\varepsilon > 0\), there exists an \(N\) such that for all \(n, m > N\), we have \(\|f_n - f_m\|\_\infty < \varepsilon\). As our space is with complex values, we can use the absolute values and triangle inequality to get \(\sup_{x\in S}|f_n(x) - f_m(x)| < \varepsilon\). From here we know that for each \(x\in S\), \((f_n(x))\) is a Cauchy sequence in the complex plane, and thus must converge to a limit, say, \(f(x)\). We can see that \(f\in F(S)\) and that \((f_n)\) converges to \(f\) under the supremum norm, hence \(F(S)\) is complete.

Show that \(\ell_{\infty}\) is a Banach space

Now, to prove that \(\ell_{\infty}\) is a Banach space, we need to show that it is complete under the supremum norm. In fact, \(\ell_{\infty}\) is a particular case of the function space \(F(S)\): it is isometrically isomorphic to the space of bounded functions, \(F(\mathbb{N})\), with supremum norm. Since we have proven that \(F(S)\) is complete with respect to the supremum norm, and isomorphisms preserve completeness, this implies that \(\ell_{\infty}\) is also complete under the supremum norm, i.e., it is a Banach space. Hence, \(\ell_{\infty}\) is a Banach space under the norm \(\|\mathrm{x}\|=\sup \left(x_{t}\right)\).

Key concepts

Supremum Norm

The supremum norm is an essential concept in functional analysis. It is a type of infinity norm, which is often used to determine the "size" of a function or sequence. The supremum norm of a function \( f \), denoted as \( \|f\|_\infty \), is defined as the supremum (or least upper bound) of the absolute values of \( f \)'s elements.
For a function over a set \( S \), this is expressed as \( \|f\|_\infty = \sup_{x \in S} \|f(x)\| \). This norm provides a way to measure how large the values of \( f \) can become, using the largest absolute value taken by \( f \) across its domain.

• It captures the behavior of functions near their "peaks," focusing on the largest deviations from zero.
• In spaces of bounded functions, like \( \ell_{\infty} \) (the space of bounded infinite sequences), the supremum norm plays a crucial role in defining the boundedness property.

Understanding and utilizing the supremum norm is pivotal in working with function spaces and vector spaces like \( \ell_{\infty} \).

Cauchy Sequence

A Cauchy sequence is a sequence whose elements become arbitrarily close to each other as the sequence progresses. In a more formal sense, a sequence \( (x_n) \) is a Cauchy sequence if, for every positive real number \( \varepsilon \), there exists an integer \( N \) such that for all integers \( m, n > N \), the distance between \( x_n \) and \( x_m \) is less than \( \varepsilon \).
This concept is crucial for defining the completeness of a space, such as in Banach spaces. In the case of a supremum norm, the Cauchy sequence \( (f_n) \) in a space \( F(S) \) will satisfy \( \sup_{x \in S} |f_n(x) - f_m(x)| < \varepsilon \), indicating that the functions \( f_n \) become increasingly similar as the sequence progresses.

• Every convergent sequence is a Cauchy sequence.
• But not every Cauchy sequence necessarily converges unless the space is complete.

This property helps in proving that function spaces and vector spaces like \( \ell_{\infty} \) are indeed complete spaces when defined with the supremum norm.

Complete Space

Complete space is a term in mathematics used to describe a space where every Cauchy sequence has a limit that is also within the space. This property is significant as it ensures all sequences that start to "settle down" eventually have endpoints within the space, offering a form of closure for analysis.
In the context of the supremum norm, proving that a space is complete involves showing that its Cauchy sequences converge within it. For instance, in the function space \( F(S) \), a Cauchy sequence \( (f_n) \) must converge to a function \( f(x) \) that is also in \( F(S) \), ensuring the space's completeness.

• A complete space under a norm is termed a Banach space.
• The completeness of function spaces like \( \ell_{\infty} \) under the supremum norm reinforces their structure as Banach spaces.

This essential property allows mathematicians to perform various types of analysis and operations knowing the integrity of the space is maintained.

Isometrically Isomorphic

An isometrically isomorphic relationship between spaces means there is a correspondence, or mapping, that not only pairs elements from one space to another but also preserves distances. This kind of mapping maintains both the structure and the "shape" of spaces in terms of their metric (measure of distance).
When considering spaces like \( \ell_{\infty} \) and a function space \( F(S) \), showing that they are isometrically isomorphic means there is an exact "mapping over" that keeps their respective supremum norms intact.

• The concept ensures that properties like completeness and boundedness are preserved across the spaces.
• Isometric isomorphisms provide a powerful tool for analyzing and comparing different mathematical structures.

This relationship is vital in proving completeness and thus categorizing spaces as Banach spaces.