Show that the set of all sequences \((x_1, x_2, \ldots)\) such that \(\sum x_i^2\) converges is complete in the \(\ell^2\)-metric. (See E-39RRX.)
Solution
Let \(\ell^2 = \{ \mathbf{x} = (x_i)_{i=1}^\infty \mid \sum_{i=1}^\infty x_i^2 < \infty \}\) with the metric \(d(\mathbf{x}, \mathbf{y}) = \left(\sum_{i=1}^\infty (x_i - y_i)^2\right)^{1/2}\).
1. Setting up Cauchy sequence and coordinate convergence:Proof:
1.1. Let \((\mathbf{x}_n)_{n=1}^\infty\) be a Cauchy sequence in \(\ell^2\), where \(\mathbf{x}_n = (x_{n, i})_{i=1}^\infty\).
1.2. For any \(\varepsilon > 0\), there exists an integer \(N \ge 1\) such that:\[d(\mathbf{x}_n, \mathbf{x}_m)^2 = \sum_{i=1}^\infty (x_{n, i} - x_{m, i})^2 < \varepsilon^2 \quad \text{for all } n, m \ge N.\]
1.3. For each fixed index \(i \ge 1\), \(|x_{n, i} - x_{m, i}|^2 \le d(\mathbf{x}_n, \mathbf{x}_m)^2 < \varepsilon^2\), so \(|x_{n, i} - x_{m, i}| < \varepsilon\).
1.4. Thus \((x_{n, i})_{n=1}^\infty\) is a Cauchy sequence in \(\mathbb{R}\).
1.5. Completeness of \(\mathbb{R}\) ensures that each coordinate sequence converges to a limit \(y_i = \lim_{n \to \infty} x_{n, i} \in \mathbb{R}\).
1.6. Define the candidate sequence \(\mathbf{y} = (y_i)_{i=1}^\infty\).
2. Proof of convergence \(\mathbf{x}_n \to \mathbf{y}\) in the \(\ell^2\) norm:Proof:
2.1. For any fixed positive integer \(K \ge 1\) and all \(n, m \ge N\):\[\sum_{i=1}^K (x_{n, i} - x_{m, i})^2 \le \sum_{i=1}^\infty (x_{n, i} - x_{m, i})^2 < \varepsilon^2.\]
2.2. Taking the limit as \(m \to \infty\) inside the finite sum:\[\sum_{i=1}^K (x_{n, i} - y_i)^2 \le \varepsilon^2 \quad \text{for all } n \ge N.\]
2.3. Because this holds for all \(K \ge 1\), taking \(K \to \infty\) gives:\[\sum_{i=1}^\infty (x_{n, i} - y_i)^2 \le \varepsilon^2 \quad \text{for all } n \ge N.\]
2.4. Thus \(\mathbf{x}_n - \mathbf{y} \in \ell^2\) and \(d(\mathbf{x}_n, \mathbf{y}) \le \varepsilon\) for all \(n \ge N\).
3. Proof that \(\mathbf{y} \in \ell^2\):Proof:
3.1. Fix \(N\) corresponding to \(\varepsilon = 1\).
3.2. By Minkowski’s inequality (triangle inequality in \(\ell^2\), E-39RRX):\[\|\mathbf{y}\|_2 = \|\mathbf{x}_N + (\mathbf{y} - \mathbf{x}_N)\|_2 \le \|\mathbf{x}_N\|_2 + \|\mathbf{y} - \mathbf{x}_N\|_2 \le \|\mathbf{x}_N\|_2 + 1 < \infty.\]
3.3. Therefore \(\mathbf{y} \in \ell^2\).
4. Q.E.D.By steps 2 and 3, the Cauchy sequence \((\mathbf{x}_n)\) converges to \(\mathbf{y} \in \ell^2\).