Let $X=\ell^2=\{(\xi_i)_{i\in\mathbb{N}}\in\mathbb{R}^\mathbb{N}\, | \, \sum_{i\in\mathbb{N}}|\xi_i|^2<\infty\}$ be the real Hilbert space of square-summable sequences, and for every $j\in\mathbb{N}$ set
$$ X_j=\{(\xi_i)_{i\in\mathbb{N}}\in X\, | \, \forall i\neq j, \xi_i=0\}.$$
It follows that $(X_j)_{j\in \mathbb{N}}$ are closed mutually orthogonal vector subspaces of $X$, and hence they are also real Hilbert spaces in their own right using the subspace topology inherited from $\ell^2$ (same norm, inner product, etc). Now, consider the "direct sum" of the $X_i$s: $(\hat{X}, \langle\cdot\, |\, \cdot \rangle)$ where
$$\hat{X}=\{ (x_j)_{j\in \mathbb{N}} \in \times_{j\in\mathbb{N}}X_j \, | \,\|(x_j)_{j\in J}\|^2_{\hat{X}}= \sum_{j\in\mathbb{N}} \|x_j\|^2_{\ell^2} < +\infty\}.\tag{1}$$
and
$$\langle (x_j)_{j\in J} \, | (y_j)_{j\in J} \rangle = \sum_{j\in\mathbb{N}} \langle x_j \, | \, y_j\rangle_{\ell_2}.\tag{2}$$
Is $(\hat{X}, \langle\cdot\, |\, \cdot \rangle)$ a valid real Hilbert space?
My colleague claims that it is not: The elements of $\hat{X}$ are a countable product of countable sequences, so in a sense it is impossible to sum them formally. Essentially they're saying that we can not append a sequence onto another sequence countably many times as done in (1). However, I have seen this direct sum construction in several books, and proofs showing it is a valid Hilbert space do not rely on the structure of the underlying $X_i$'s at all (other than the fact that they are real Hilbert spaces to start with). What do you think?
Best Answer
I guess you have a typo and want to have
$$\hat X = \{(x_j) \mid \sum_j \lVert x_j \rVert^2_{\ell^2} < \infty \}.$$ Then $\hat X$ is a Hilbert space. Your construction works for any collection of Hilbert spaces $X_j$, not only for the subspaces $X_j \subset \ell^2$ defined in your question.
See for example section "Direct sums" in https://en.wikipedia.org/wiki/Hilbert_space#Direct_sums.
Note that if you define $$\tilde X = \{(x_j) \mid \sum_j \lVert x_j \rVert_{\ell^2} < \infty \},$$ then $\tilde X \subset \hat X$, thus your definition $$\langle (x_j), (y_j) \rangle = \sum_{j\in\mathbb{N}} \langle x_j , y_j\rangle_{\ell_2}$$ defines an inner product on $\tilde X$. However, $\tilde X$ is not a Hilbert space - it lacks completeness.
By the way, in your example the $X_j$ are one-dimensional subpaces of $\ell^2$ and therefore isomorphic to $\mathbb R$. Thus $\hat X$ is nothing else than a copy of $\ell^2$.