Consider two Hilbert spaces $H_1$ and $H_2$ with inner products $\langle \cdot,\cdot\rangle_1$ and $\langle \cdot,\cdot\rangle_2$ generating norms $\Vert \cdot \Vert_1$ and $\Vert \cdot \Vert_2$ respectively. I am trying to understand when $H = H_1 \cap H_2$ becomes a Hilbert space with the inner product $\langle \cdot,\cdot\rangle_1 + \langle \cdot,\cdot\rangle_2$. Or does this always hold? I can see that a Cauchy sequence in $H$ is Cauchy in both $H_1$ and $H_2$, but apriori, it is not clear to me that a common convergent subsequence can be found. Thanks for any help!
Edit: Let us assume that $H_1$ and $H_2$ are sitting inside a larger Hilbert space $H$. As pointed out below, otherwise the question doesn't make sense.