Banach space is a complete vector space, on which a norm \({\lVert \; \rVert }\) is defined.
- A distance between vectors:
$$d(v,w) =\lVert v-w \rVert$$
- Every Cauchy sequence of vectors always converges to a limit that is also in that space.

Hilbert space is a complete vector space on which there is an inner product \({\langle \;, \;\rangle }\).
- A distance between vectors:
$$d(v,w) =\lVert v-w \rVert=\sqrt{\langle v-w, v-w\rangle}$$
- Every Cauchy sequence of vectors always converges to a limit that is also in that space.

Square-integrable functions space \(L^2\)
$$f: \mathrm{R} \rightarrow \mathrm{C}$$
Inner product
$$ \langle f , g \rangle =\int_{-\infty} ^{\infty} f^{*} (x) g(x) dx$$
Norm
$$ \lVert f \rVert^2=\int_{-\infty} ^{\infty} |{f(x)} |^2 dx<\infty$$