is this true for Hilbert space direct sum of $H$ when $H$ is infinite dimensional?

Question

Let $(H_{\alpha})_{{\alpha \in I}}$ be a $I-$indexed family of Hilbert spaces over $\mathbb{F}$.

let $H=\bigoplus H_\alpha$ be their Hilbert space direct sum. Can we say $\dim H=\sum\limits_{\alpha\in I} \dim H_\alpha$ if H is infinite dimensional?

thanks for your help

Answer 1

Yes we can say this for all no matter what $\operatorname{hilb.dim}(H)$ is. Let $S_i$ be a set of cardinality $\operatorname{hilb.dim}(H_i)$, then $H_i=\ell_2(S_i)$. From this post we know that $$ H=\bigoplus_{i\in I} H_i=\bigoplus_{i\in I}\ell_2(S_i)=\ell_2\left(\bigsqcup_{i\in I}S_i\right) $$ Hence $$ \operatorname{hilb.dim}(H)=\operatorname{Card}\left(\bigsqcup_{i\in I}S_i\right)=\sum_{i\in I}\operatorname{Card}(S_i)=\sum_{i\in I}\operatorname{hilb.dim}(H_i) $$