# If a Hilbert space is linearly isometric to a Banach space, will the Banach space be a Hilbert space? [duplicate]

Question: Given two Banach spaces $$X$$ and $$Y$$, if there exists an onto linear isometry $$T:X\to Y$$ and $$X$$ is a Hilbert space, is it true that $$Y$$ is a Hilbert space?

Intuitively this seems true to me. To prove this, one just need to construct an $$\langle\cdot, \cdot \rangle$$ on $$Y\times Y$$ and show that it is an inner product.