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A Hilbert space $H$ is a real or complex vector space endowed with an inner product such that $H$ is a complete metric space when endowed with the norm induced by this inner product.

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Interpolation of embedded Hilbert spaces and intersection

I'm wondering under what hypothesis it is true a property like $$[\mathcal{H}_1\cap X, \mathcal{H}_2\cap X]_{\theta}=\mathcal{H}_1\cap X\cap [\mathcal{H}_1, \mathcal{H_2}]_{\theta}$$ where $\mathcal{H …
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