I am a bit confused about the topological relation between the direct sum and direct product of separable (possibly infinite dimensional) Hilbert spaces $\{H_n\}$.
In order to avoid complications further, suppose that $n$ is a countable index.
Then, we can first think of the 'direct product' of $\{H_n\}$, denoted as \begin{equation} \prod_{n=1}^\infty H_n \end{equation} equipped with the product topology.
And the 'orthogonal' direct sum of $\{H_n\}$ is a vector subspace of $\prod_{n=1}^\infty H_n$ defined by \begin{equation} \bigoplus_{n=1}^\infty H_n :=\Bigl\{ (x_n) \in \prod_{n=1}^\infty H_n \mid \sum_{n=1}^\infty \lVert x_n \rVert^2 < \infty \Bigr\}, \end{equation} which is a Hilbert space itself.
However, I am not sure whether $\bigoplus_{n=1}^\infty H_n$ is a 'topological subspace' of $\prod_{n=1}^\infty H_n$. That is, I wonder if \begin{equation} \text{the topology of }\bigoplus_{n=1}^\infty H_n\text{ inherited from }\prod_{n=1}^\infty H_n \end{equation} is equal to the Hilbert space topology of $\bigoplus_{n=1}^\infty H_n$.
I think the answer is yes, but cannot figure out how to start a rigorous proof.
Could anyone please help me?