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--- hilbert-space [May 09, 2026 at 02:06] – yanevskiv
+++ hilbert-space [May 14, 2026 at 11:38] (current) – external edit 127.0.0.1
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+# Hilbert space
+**Hilbert space** is a complete inner product space. It is vector space that is equipped with an inner product such that the metric induced by the inner product is complete within the space. Completeness means that all Cauchy sequences converge within the same space. Hilbert space is a special case of the more general Banach space, both of which are concepts from functional analysis.
+What does it mean Think about it. Where do real numbers live? They live on the real number line $\mathbb{R}$. Where do complex numbers live? They live on the complex plane $\mathbb{C}$. Where do quantum states $\lvert\psi\rangle$ live? They live in a Hilbert space.