quantum mechanics - Direct Sum of Hilbert spaces

I am a physicist who is not that well-versed in mathematical rigour (a shame, I know! But I'm working on it.) In Wald's book on QFT in Curved spacetimes, I found the following definitions of the direct sum of Hilbert spaces. He says -

Next, we define the direct sum of Hilbert spaces. Let $\{ {\cal H}_\alpha \}$ be an arbitrary collection of Hilbert spaces, indexed by $\alpha$ (We will be interested only with the case where there are at most a countable number of Hilbert spaces, but no such restriction need be made for this construction). The elements of the Cartesian product $\times_\alpha {\cal H}_\alpha$ consist of the collection of vectors $\{\Psi_\alpha\}$ for each $\Psi_\alpha \in {\cal H}_\alpha$. Consider, now, the subset, $V \subset \times_\alpha {\cal H}_\alpha$, composed of elements for which all but finitely many of the $\Psi_\alpha$ vanish. Then $V$ has the natural structure of an inner product space. We define the direct sum Hilbert space $\bigoplus\limits_\alpha {\cal H}_\alpha$ to be the Hilbert space completion of $V$. It follows that in the case of a countable infinite collection of Hilbert spaces $\{{\cal H}_i\}$ each $\Psi \in \bigoplus\limits_i {\cal H}_i$ consists of arbitrary sequences $\{\Psi_i\}$ such that each $\Psi_i \in {\cal H}_i$ and $\sum\limits_i \left\| \Psi_i \right\|_i^2 < \infty$.

Here $\left\| ~~\right\|_i$ is the norm defined in ${\cal H}_i$. Also, Hilbert space completion of an inner product vector space $V$ is a space ${\cal H}$ such that $V \subset {\cal H}$ and ${\cal H}$ is complete in the associated norm. It is constructed from $V$ by taking equivalence classes of Cauchy sequences in $V$.

Now the questions -

1. Why does $V$ have the structure of an inner product space?

2. How does he conclude that $\sum\limits_i \left\| \Psi_i \right\|_i^2 < \infty$?

3. How does this definition of the direct sum match up with the usual things we see when looking at tensors in general relativity or in representations of Lie algebras, etc.?

PS - I also have a similar problem with Wald's definition of a Tensor Product of Hilbert spaces. I have decided to put that into a separate question. If you could answer this one, please consider checking out that one too. It can be found here. Thanks!

Answer

An element of the direct sum $H_1\oplus H_2\oplus ...$ is a sequence $$\Psi = \{\Psi_1, \Psi_2,...\}$$ consisting of an element from $H_1$, and element from $H_2$...etc (countable). This has to have the special property that the sum $$\left\| \Psi_1\right\|^2+\left\| \Psi_2\right\|^2+... $$converges. This convergence is part of the definition of direct sum.

To show that the direct sum is a Hilbert space, I need well defined addition operation. If I have a pair of such things: $$ \Psi = \{\Psi_1, \Psi_2,...\}$$ $$ \Phi = \{\Phi_1, \Phi_2,...\}$$ I can define their sum to be the sequence $$ \Phi+\Psi = \{\Phi_1+\Psi_1, \Phi_2+\Psi_2,...\}$$We need to check that $$\left\| \Psi_1+\Phi_1 \right\|^2+\left\| \Psi_2+\Phi_2 \right\|^2+...$$converges. To do this, we use the fact that the individual terms satisfy $$\left\| \Psi_i+\Phi_i \right\|^2 $$ $$=\left\| \Psi_i\right\|^2+\left\| \Phi_i\right\|^2+(\Psi_i,\Phi_i)+(\Phi_i,\Psi_i) $$ $$ \leq \left\| \Psi_i\right\|^2+\left\| \Phi_i\right\|^2+2\left\| \Psi_i\right\|\left\| \Phi_i\right\| $$ $$ \leq 2\left\| \Psi_i\right\|^2+2\left\| \Phi_i\right\|^2$$So convergence follows from the convergence properties of the individual spaces being summed. So this shows how we can add elements of the direct sum. Scalar products follow in the same way.

To define an inner product, we just add the inner products component wise, i.e $$(\Psi, \Phi) = (\Psi_1,\Phi_1)+(\Psi_2,\Phi_2)+... \ \ \ (1)$$To show the RHS converges, note $$|(\Psi_i, \Phi_i)| \leq \left\| \Psi_i \right\| \left\| \Phi_i \right\| $$But $$2\left\| \Psi_i \right\| \left\| \Phi_i \right\| \leq \left\| \Psi_i \right\|^2+ \left\| \Phi_i \right\|^2$$so the RHS of (1) converges absolutely and we are done - we have a well defined inner product space.