Show that $l^2$ is a Hilbert space

In this answer, I will use $x_n$ as a sequence in $l^2$ and write $x_n(k)$ as the $k$-th member of that sequence.

The norm in the Hilbert space is given by $\|x\| = \sqrt{\langle x, x \rangle}$. We wish to show that if a sequence $\{ x_n \} \subset l^2$ is Cauchy, then it converges in $l^2$.

Suppose that $\{x_n\}$ is such a Cauchy sequence. Let $\{ e_k \}$ be the collection of sequences for which $e_k(i) = 1$ if $i=k$ and zero if $i\neq k$.

Then $\langle x_n, e_k \rangle = x_n(k)$. Notice that $$|x_n(k) - x_m(k)| = |\langle x_n - x_m, e_k \rangle| \le \|x_n-x_m\| \| e_k\| = \|x_n-x_m\|$$ for all $k$ (also note that this convergence is uniform over $k$). Therefore the sequence of real numbers given by $\{x_n(k)\}_{n\in \mathbb{N}}$ is Cauchy for each $k$, and thus converges. Call the limit of this sequence $\tilde x(k)$.

Let $\tilde x = (\tilde x(k))_{k\in\mathbb{N}}$. We wish to show that $\tilde x \in l^2$.

Consider $$\sum_{k=1}^\infty |\tilde x(k)|^2=\sum_{k=1}^\infty |\lim_{n\to\infty} x_n(k)|^2=\lim_{n\to\infty} \sum_{k=1}^\infty |x_n(k)|^2=\lim_{n\to\infty}\|x_n\|^2.$$

The exchange of limits is justified, since the convergence of $\lim_{n\to\infty} x_n(k)$ is uniform over $k$. Finally, since $\{ x_n \}$ is Cauchy, the inequality, $$| \|x_m\| - \|x_n\| | < \| x_m - x_n\|$$ implies that $\|x_n\|$ is a Cauchy sequence of real numbers, and so $\|x_n\|$ converges. Thus $\tilde x$ is in $l^2$.

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Edit: Completing the proof as per the comments.

We have thus shown that $\tilde x$ is in $l^2$. $\tilde x$ is the most likely candidate for the Cauchy sequence to converge to, and it has been demonstrated to be in our space. What remains is to show that $$\| x_n - \tilde x\| \to 0$$ as $n \to \infty$.

We will utilize a generalized form of the dominated convergence theorem for series. This states that if $a_{n,k} \to b_k$ for all $k$, $a_{n,k} < d_{n,k}$ and $\sum_{k} d_{n,k} \to \sum_{k} D_k < \infty$, then $\lim_{n \to \infty} \sum_{k=0}^\infty a_{n,k} = \sum_{k=0}^\infty b_k$. (here $a_{n,k}, b_k, d_{n,k}, D_{k}$ are all real numbers)

Writing $$\| x_n - \tilde x\|^2 = \sum_{k=0}^\infty |x_n(k) - \tilde x(k)|^2.$$

We see that in this case $a_{n,k} = |x_{n}(k) - \tilde x(k)|^2$, $b_k = 0$, and we must find a $d_{n,k}$ that "dominates" $a_{n,k}$ to finish the proof.

Now note that $|x_n(k) - \tilde x(k)|^2 \le 2 |x_n(k)|^2 + 2 |\tilde x(k)|^2$ and $$\lim_{n \to \infty} \sum_{n=0}^\infty ( 2 |x_n(k)|^2 + 2 |\tilde x(k)|^2) = \sum_{k=0}^\infty (2 |\tilde x(k)|^2 + 2 | \tilde x(k)|^2).$$ Recall that we demonstrated $\lim_{n \to \infty} \sum_{n=0}^\infty |x_n(k)|^2 = \sum_{n=0}^\infty |\tilde x(k)|^2$ in the first half. Thus $D_k$ is played by $4|\tilde x(k)|^2$ in this case.

Thus by the dominated convergence theorem we may conclude that $$\sum_{k=0}^\infty |x_n(k)-\tilde x(k)|^2 \to 0.$$

Let $(\mathbf{x_n})$ be a Cauchy sequence in $l^2$, where $\mathbf{x_n} = (x_1^{(n)},x_2^{(n)},\ldots)$, i.e., given $\epsilon >0$ there exists a natural number $N$ such that for all $m,n\geq N$ \begin{equation} \|\mathbf{x_n}-\mathbf{x_m}\| = \left(\sum_\limits{j=1}^{\infty}|x_j^{(n)}-x_j^{(m})|^2\right)^{\frac{1}{2}} <\epsilon \end{equation} In particular, it follows that for every $j=1,2,\ldots$ we have \begin{align} |x_j^{(n)}-x_j^{(m)}| < \epsilon && (m,n\geq N). \end{align} That is for each fixed $j$ the sequence $(x_j^{(n)},x_j^{(n)},\ldots)$ is a Cauchy sequence in the scalar field $\mathbb{R}$ or $\mathbb{C}$ and hence it converges. Let $x_j^{(n)} \to x_j$ as $n \to \infty$. Using these limits now define $\mathbf{x} = (x_1,x_2,\ldots)$. Now, with this basic setting try to show that $(\mathbf{x_n})$ converges to $\mathbf{x}$.

A typical proof of the completeness of $\ell^2$ consists of two parts.

Reduction to series

Claim: Suppose $ X$ is a normed space in which every absolutely convergent series converges; that is, $ \sum_{n=1}^{\infty} y_n$ converges whenever $ y_n\in X$ are such that $ \sum_{n=1}^{\infty} \|y_n\|$ converges. Then the space $X$ is complete.

Proof. Take a Cauchy sequence $ \{x_n\}$ in $X$. For $ j=1,2,\dots$ find an integer $ n_j$ such that $ \|x_n-x_m\|<2^{-j}$ as long as $ n,m\ge n_j$. (This is possible because the sequence is Cauchy.) Also let $ n_0=1$ and consider the series $ \sum_{j=1}^{\infty} (x_{n_{j}}-x_{n_{j-1}})$. This series converges absolutely, by comparison with $\sum 2^{-j}$. Hence it converges. Its partial sums simplify (telescope) to $ x_{n_j}-x_1$. It follows that the subsequence $ \{x_{n_j}\}$ has a limit. It remains to apply a general theorem about metric spaces: if a Cauchy sequence has a convergent subsequence, then the entire sequence converges. $\quad \Box$

Convergence of absolutely convergent series in $\ell^2$

Claim:: Every absolutely convergent series in $ \ell^2$ converges

Proof. The elements of $ \ell^2$ are functions from $ \mathbb N$ to $ \mathbb C$, so let's write them as such: $ f_j: \mathbb N\to \mathbb C$. (This avoids confusion of indices.) Suppose the series $ \sum_{j=1}^{\infty} \|f_j\|$ converges. Then for any $ n$ the series $ \sum_{j=1}^{\infty} f_j(n)$ converges, by virtue of comparison $|f_j(n)| \le \|f_j\|$.

Let $ f(n) = \sum_{j=1}^{\infty} f_j(n)$. So far the convergence is only pointwise, so we are not done. We still have to show that the series converges in $ \ell^2$, that is, its tails have small $ \ell^2$ norm: $ \sum_{n=1}^\infty |\sum_{j=k}^{\infty} f_j(n)|^2 \to 0$ as $ k\to\infty$.

What we need now is a dominating function/sequence (sequences are just functions with domain $\mathbb{N}$), in order to apply the Dominated Convergence Theorem. Namely, we need a function $ g: \mathbb N\to [0,\infty)$ such that

$$ \sum_{n=1}^{\infty} g(n)^2<\infty \tag{1}$$ $$ \left|\sum_{j=k}^{\infty} f_j(n)\right| \le g(n) \quad \text{for all } \ k,n \tag{2} $$

Set $ g(n) = \sum_{j=1}^{\infty} |f_j(n)| $. Then (2) follows from the triangle inequality. Also, $ g$ is the increasing limit of functions $ g_k(n) = \sum_{j=1}^k |f_j(n)| $. For each $k$, using the triangle inequality in $ \ell^2$, we get $$ \sum_n g_k(n)^2 = \|g_k\|^2\le \left(\sum_{j=1}^k \|f_j\|\right)^2 \le S^2$$ where $S= \sum_{j=1}^\infty\|f_j\|$. Therefore, $ \sum_n g(n)^2\le S^2$ by the Monotone Convergence Theorem.

To summarize: we have shown the existence of a summable function $g^2$ that dominates the square of any tail of the series $\sum f_j$. This together with the pointwise convergence of said series yield its convergence in $\ell^2$. $\quad\Box$