Question

(a) Show that \(\mathbb{R}_{p}^{n}\) is complete if \(p \geq 1\). (b) Show that \(\ell_{p}\) is complete if \(p \geq 1\).

Short answer

Question: Prove that the metric spaces of (a) \(n\)-dimensional real vectors with respect to the p-norm, \(\mathbb{R}_{p}^{n}\), and (b) the infinite sequences of real numbers with finite p-norm, \(\ell_{p}\), are complete when \(p \geq 1\). Answer: To show that these spaces are complete, we have demonstrated that every Cauchy sequence in the respective spaces converges to a limit within the space. We have done this for both (a) \(\mathbb{R}_{p}^{n}\) and (b) \(\ell_{p}\). (a) We showed that a Cauchy sequence in \(\mathbb{R}_{p}^{n}\), denoted as \(\{\mathbf{x}_k\}_{k=1}^{\infty}\), converges to a limit vector \(\mathbf{x} \in \mathbb{R}_{p}^{n}\) by proving convergence of its component sequences and then using the triangle inequality for the p-norm. (b) Similarly, we showed that a Cauchy sequence in \(\ell_{p}\), denoted as \(\{\mathbf{x}_k\}_{k=1}^{\infty}\), converges to a limit sequence \(\mathbf{x} \in \ell_{p}\) by proving convergence of its component sequences and then using the triangle inequality for the p-norm. Therefore, both metric spaces (\(\mathbb{R}_{p}^{n}\) and \(\ell_{p}\)) are complete when \(p \geq 1\).

Step-by-step solution

Define Cauchy Sequence in \(\mathbb{R}_{p}^{n}\)

Let \(\{\mathbf{x}_k\}_{k=1}^{\infty}\) be a Cauchy sequence in \(\mathbb{R}_{p}^{n}\), where \(\mathbf{x}_k = (x_{k1}, x_{k2}, \dots, x_{kn})\). This means that for any \(\epsilon > 0\), there exists a positive integer \(N\) such that for all \(m, n > N\), \(||\mathbf{x}_m - \mathbf{x}_n||_p < \epsilon\).

Show Convergence of Component Sequences

For each ith component, we have the sequence of real numbers \(\{x_{ki}\}_{k=1}^{\infty}\). Since the real line is complete, each of these sequences will converge: for each \(i\), \(\lim_{k \to \infty} x_{ki} = x_i\). Let the limit vector \(\mathbf{x} = (x_1, x_2, \dots, x_n)\).

Proving Convergence of the Cauchy Sequence

Now, we will show that \(\lim_{k \to \infty} ||\mathbf{x}_k - \mathbf{x}||_p = 0\). Using the triangle inequality for the p-norm and that \(\{\mathbf{x}_k\}_{k=1}^{\infty}\) is a Cauchy sequence, we have: \(key_placeholder\)||\mathbf{x}_k - \mathbf{x}||_p = \lim_{m \to \infty}||\mathbf{x}_k - \mathbf{x}_m + \mathbf{x}_m - \mathbf{x}||_p \leq \lim_{m \to \infty}||\mathbf{x}_k - \mathbf{x}_m||_p + ||\mathbf{x}_m - \mathbf{x}||_p \leq \epsilon + \epsilon = 2\epsilon\( when \)m, k > N$. As \(\epsilon\) can be made arbitrarily small, the limit converges, meaning that \(\lim_{k \to \infty} ||\mathbf{x}_k - \mathbf{x}||_p = 0\). Thus, the Cauchy sequence \(\{\mathbf{x}_k\}_{k=1}^{\infty}\) converges to the vector \(\mathbf{x} \in \mathbb{R}_{p}^{n}\). (b) Show that \(\ell_{p}\) is complete if \(p \geq 1\).

Define Cauchy Sequence in \(\ell_{p}\)

Let \(\{\mathbf{x}_k\}_{k=1}^{\infty}\) be a Cauchy sequence in \(\ell_{p}\), where \(\mathbf{x}_k = (x_{k1}, x_{k2}, \dots)\). This means that for any \(\epsilon > 0\), there exists a positive integer \(N\) such that for all \(m, n > N\), \(||\mathbf{x}_m - \mathbf{x}_n||_p < \epsilon\).

Show Convergence of Component Sequences

Once again, for each ith component, we have the sequence of real numbers \(\{{x_{ki}}\}_{k=1}^{\infty}\). Since the real line is complete, each of these sequences will converge: \(\lim_{k \to \infty} {x_{ki}} = x_i\). Let the limit sequence \(\mathbf{x} = (x_1, x_2, \dots)\).

Proving Convergence of the Cauchy Sequence

Now, we will show that \(\lim_{k \to \infty} ||\mathbf{x}_k - \mathbf{x}||_p = 0\). Using the triangle inequality for the p-norm, we have: \(key_placeholder\)||\mathbf{x}_k - \mathbf{x}||_p = \lim_{m \to \infty}||\mathbf{x}_k - \mathbf{x}_m + \mathbf{x}_m - \mathbf{x}||_p \leq \lim_{m \to \infty}||\mathbf{x}_k - \mathbf{x}_m||_p + ||\mathbf{x}_m - \mathbf{x}||_p \leq \epsilon + \epsilon = 2\epsilon\( when \)m, k > N$. As \(\epsilon\) can be made arbitrarily small, the limit converges, meaning that \(\lim_{k \to \infty} ||\mathbf{x}_k - \mathbf{x}||_p = 0\). Thus, the Cauchy sequence \(\{\mathbf{x}_k\}_{k=1}^{\infty}\) converges to the sequence \(\mathbf{x} \in \ell_{p}\). Therefore, \(\ell_{p}\) is also complete when \(p \geq 1\).

Key concepts

Cauchy Sequences

A Cauchy sequence is a sequence whose elements become arbitrarily close to each other as the sequence progresses. In formal terms, a sequence \( \{x_k\}_{k=1}^{\infty} \) is called a Cauchy sequence if for every positive number \( \epsilon > 0 \) there exists an integer \( N \) such that for all \( m, n > N \) the distance between \( x_m \) and \( x_n \) is less than \( \epsilon \)—that is, \( |x_m - x_n| < \epsilon \).
In real analysis, the idea of a Cauchy sequence is critical because it provides a way to define convergence in spaces that may not have a standard notion of distance. For instance, in the exercise above, proving that a particular space is complete involves showing that every Cauchy sequence in that space converges to a limit within the space. For spaces \( \mathbb{R}_{p}^{n} \) and \( \ell_{p} \) with \( p \geq 1 \), the completeness is shown by demonstrating that any Cauchy sequence has a limit that is also within the respective spaces.

p-norm

The p-norm is a generalization of the familiar Euclidean norm and is defined for sequences and vectors. For a given vector \( \mathbf{x} = (x_1, x_2, \dots, x_n) \) in \( \mathbb{R}^{n} \) and a real number \( p \geq 1 \), the p-norm (also known as \( L^p \) norm) is given by \( ||\mathbf{x}||_p = (\sum_{i=1}^{n} |x_i|^p)^{1/p} \).
For sequences, the same principle applies; however, since sequences can be infinite, the p-norm takes the form of an infinite series: \( ||\mathbf{x}||_p = (\sum_{i=1}^{\infty} |x_i|^p)^{1/p} \), provided the series converges. The p-norm is a crucial concept in real analysis and functional analysis because it allows for the measurement of 'lengths' and distances in more generalized vector spaces and sequence spaces like \( \mathbb{R}_{p}^{n} \) and \( \ell_{p} \).

Convergence of Sequences

Convergence of sequences is a fundamental concept in real analysis and it describes the behavior of sequences approaching a limit. A sequence \( \{x_k\}_{k=1}^{\infty} \) converges to a limit \( L \) if, as \( k \) becomes large, the terms \( x_k \) get arbitrarily close to \( L \)—meaning for every \( \epsilon > 0 \) there exists an integer \( N \) such that for all \( k > N \) we have \( |x_k - L| < \epsilon \).
The exercise provided required demonstrating that each component sequence of a Cauchy sequence in \( \mathbb{R}_{p}^{n} \) or \( \ell_{p} \) converges, which leverages the completeness of the real numbers. Convergence of sequences is especially important in the context of Cauchy sequences when discussing completeness, since a space is complete if and only if every Cauchy sequence in it converges to an element within the space.

Triangle Inequality

The triangle inequality is a pivotal inequality in mathematics, stating that for any vectors \( \mathbf{x} \) and \( \mathbf{y} \) in a normed vector space, the length of the sum of the vectors is no greater than the sum of the lengths of the vectors: \( ||\mathbf{x} + \mathbf{y}|| \leq ||\mathbf{x}|| + ||\mathbf{y}|| \).
This concept is utilized in multiple areas of mathematics, including the analysis of convergent sequences and the completeness of a space. In the context of the exercise, the triangle inequality is used to help establish the convergence of the Cauchy sequence to a specific limit by bounding the norm of the difference between the sequence term and the limit. This property ensures that the sequences do not 'spread out' but move closer to the limit, facilitating the proof of completeness in normed vector spaces such as \( \mathbb{R}_{p}^{n} \) and \( \ell_{p} \).