stylistic:tone, quality:corruption, information:filtering, information:classification, stylistic:style_transfer
Functions

The dot product is defined for vectors that have a finite number of entries. Thus these vectors can be regarded as discrete functions: a length- n {\displaystyle n} vector u {\displaystyle u} is, then, a function with domain { k ∈ N: 1 ≤ k ≤ n } {\displaystyle \{k\in \mathbb {N}:1\leq k\leq n\}}, and u i {\displaystyle u_{i}} is a notation for the image of i {\displaystyle i} by the function/vector u {\displaystyle u}.

This notion can be generalized to continuous functions: just as the inner product on vectors uses a sum over corresponding components, the inner product on functions is defined as an integral over some interval:

⟨ u, v ⟩ = ∫ a b u (x) v (x) d x. {\displaystyle \left\langle u,v\right\rangle =\int _{a}^{b}u(x)v(x)\,dx.}

Generalized further to complex functions ψ (x) {\displaystyle \psi (x)} and χ (x) {\displaystyle \chi (x)}, by analogy with the complex inner product above, gives

⟨ ψ, χ ⟩ = ∫ a b ψ (x) χ (x) ¯ d x. {\displaystyle \left\langle \psi,\chi \right\rangle =\int _{a}^{b}\psi (x){\overline {\chi (x)}}\,dx.}