Question

Let \(\Gamma\) be a set and \(p \in[1, \infty]\). Show that \(c_{0}(\Gamma)\) and \(\ell_{p}(\Gamma)\) are Banach spaces.

Short answer

Both \(\c_{0}(\Gamma)\) and \(\ell_{p}(\Gamma)\) are Banach spaces since every Cauchy sequence in these spaces converges within the respective space.

Step-by-step solution

- Recap the definition of a Banach space

A Banach space is a complete normed vector space. This means that every Cauchy sequence in this space converges to a limit within the space.

- Define the spaces

For a set \(Gamma\), define the spaces:\[c_{0}(Gamma) = \{x: Gamma \to \mathbb{R} \ | \ \forall \varepsilon > 0, \ \exists \text{a finite set } F \subseteq Gamma, \text{ such that } \| x_i \| < \varepsilon \forall i \otin F \}\]\[l_{p}(Gamma) = \{x: Gamma \to \mathbb{R}, \| x \|_p = \left( \sum_{i \in Gamma} \| x_i \|^p \right)^{1/p} < \infty \}\]

- Show that \(c_{0}(Gamma)\) is a Banach space

Consider a Cauchy sequence \(\{x^{(n)}\}\) in \(c_{0}(Gamma)\). By definition, for every \varepsilon > 0\, there exists \ N \in \mathbb{N} \ such that for all \ n,m \ ge \ N \, \|x^{(n)} - x^{(m)}\|_{\infty} < \varepsilon.\ Since \mathbb{R}\ is complete, the pointwise limit, say \[x_i = \lim_{n \to \infty} x^{(n)}_i,\]\ where \| x_i\| \in \mathbb{R},\ exists for each \ i \in \Gamma.\ Consequently, \{x_i\}\ is in \c_{0}(Gamma)\. Therefore, \c_{0}(Gamma)\ is complete, i.e., a Banach space.

- Show that \(\ell_{p}(\Gamma)\) is a Banach space

Consider a Cauchy sequence \{x^{(n)}\}\ in \(\ell_{p}(\Gamma)\). By definition, for any \varepsilon > 0\, there exists \ N \in \mathbb{N} \ such that for all \ n, m \ge \ N, \|x^{(n)} - x^{(m)}\|_p < \varepsilon.\ The convergence of these sequences in \(\ell_{p}(\Gamma)\) implies that the limit sequence, say \[x = (x_i) \in \ell_{p}(\Gamma)\]\ also exists pointwise since each \|x^{(n)}_i \| \in \mathbb{R},\ and: \[x_i = \lim_{n \to \infty} x^{(n)}_i.\]\ Hence, \{x_i\}\ is in \ell_{p}(Gamma).\ Therefore, \ell_{p}(Gamma)\ is complete, i.e., a Banach space.

Key concepts

Cauchy sequence

A Cauchy sequence in a normed vector space is a sequence \(\text{{x}}^{(n)}\) where, as the sequence progresses, the elements get arbitrarily close to each other. More formally, for any \(\boldsymbol{\varepsilon} > 0\), there exists an \(\boldsymbol{N}\) such that for all \(\boldsymbol{n,m \ge \text{{N}}}\), the distance between the elements \(\boldsymbol{\text{{x}}^{(n)}}\) and \(\boldsymbol{\text{{x}}^{(m)}}\) is less than \(\boldsymbol{\varepsilon}\):\(\boldsymbol{\|\text{{x}}^{(n)} - \text{{x}}^{(m)}}\| < \varepsilon\).
Cauchy sequences are important because, in a Banach space, every Cauchy sequence converges to a limit within that space. This property is what makes Banach spaces 'complete'.
Think of it this way: Imagine an infinite sequence of points getting closer and closer together on a graph. If you're in a Banach space, these points will eventually settle at a single spot.

Normed vector space

A normed vector space combines both algebraic and geometric properties. It is a vector space equipped with a norm. A norm is a function that assigns a non-negative length or size to each vector in the space.
The norm \(\boldsymbol{\|\cdot\|}\) must satisfy certain properties:

• Non-negativity: \(\boldsymbol{\|x\| \ge 0}\) for all vectors \(\boldsymbol{x}\).
• Definiteness: \(\boldsymbol{\|x\| = 0}\) if and only if \(\boldsymbol{x = 0}\).
• Scalar multiplication: \(\boldsymbol{\|\alpha x\| = \|\alpha\| \cdot \|x\|}\).
• Triangle inequality: \(\boldsymbol{\|x + y\| \le \|x\| + \|y\|}\).

A familiar example of a normed vector space is the Euclidean space \(\boldsymbol{\mathbb{R}^n}\) with the standard Euclidean norm. This simply measures the 'usual' distance of vectors in \(\boldsymbol{\mathbb{R}^n}\).

Pointwise limit

The pointwise limit of a sequence of functions refers to the value that each function in the sequence approaches at each individual point as the sequence progresses to infinity.
Formally, given a sequence of functions \(\boldsymbol{f_n}\) and a limit function \(\boldsymbol{f}\), we say \(\boldsymbol{f}\) is the pointwise limit if for each input \(\boldsymbol{x}\), \(\boldsymbol{f(x) = \lim_{n \to \infty}} f_n(x)\).
In normed vector spaces, considering sequences element-wise helps understand the convergence behavior. Each element must converge to the corresponding element in the limit function for the entire sequence to be considered convergent pointwise.

lp spaces

\(\boldsymbol{\ell_p(\Gamma)}\) spaces are a family of normed vector spaces. They consist of all functions from a set \(\boldsymbol{\Gamma}\) to the real numbers where the \(\boldsymbol{p}\)-th power of the absolute values is summable.
Specifically, \(\boldsymbol{\ell_p(\Gamma) = \{x: \Gamma \to \mathbb{R} \, | \, \|x\|_p = \( \sum_{i \, \in \, \Gamma} \| x_i \|^p \)^{1/p} < \infty \}}\) for \(\boldsymbol{1 \le p < \infty}\).
This definition highlights the importance of the norm in these spaces:

• For \(\boldsymbol{\ell_1}\)-spaces, the norm is the sum of absolute values.
• For \(\boldsymbol{\ell_2}\)-spaces, the norm is the square root of the sum of squares, which is the Euclidean norm.
• For higher \(\boldsymbol{p}\) values, the norms adjust accordingly, emphasizing different aspects of the vectors involved.

These spaces are Banach spaces because they are complete under their respective norms.

c0 spaces

\(\boldsymbol{c_0(\Gamma)}\) spaces are another important type of normed vector spaces, consisting of all functions from a set \(\boldsymbol{\Gamma}\) to the real numbers that converge to zero outside of a finite subset of \(\boldsymbol{\Gamma}\).
Formally, \(\boldsymbol{c_0(\Gamma) = \{x: \Gamma \to \mathbb{R} \, | \, \forall \varepsilon > 0 \, \exists \text{a finite set } F \subseteq \Gamma \text{such that } \| x_i \| < \varepsilon \text{ for all } i \otin \text{F} \}}\).
This means, outside of some finite set, the values of the function get arbitrarily small.
Like \(\boldsymbol{\ell_p}\)-spaces, \(\boldsymbol{c_0}\) spaces are Banach spaces because they are complete under their norm. Consider how useful these spaces can be in scenarios where functions or sequences need to 'settle down' to zero in most places, with only a few possibly non-zero values.