Intersection of Hilbert spaces

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.

Asking questions about the intersection $H_1 \cap H_2$ of two arbitrary Hilbert spaces $H_1$ and $H_2$ is "evil" in the mathematical sense of the word. Unless the two Hilbert spaces are linear subspaces of some larger Hilbert space, then this question is ill-posed.
In general, it is "evil" to ask questions about structured objects such as groups, topological spaces, etc. which depend only on the specific set-theoretic construction of that object. For example, asking if $1 \in 3$ is a bad question, because it depends on the specific set-theoretic construction of the number $1$ and the number $3$, and it may or may not be true depending on what sort of construction you have in mind.
If you consider two reproducing kernel Hilbert spaces $(H_1(X), \| \cdot\|_1)$, $(H_2(X), \| \cdot\|_2)$ on the same set $X$ then their intersection is again a reproducing kernel Hilbert space with the norm $\| \cdot \| = \sqrt{ \| \cdot\|_1^2 + \| \cdot \|_2^2}$. For more details (including definitions etc.) you may want to see this thesis http://www.thesis.bilkent.edu.tr/0002953.pdf pages 5-20.