Definition:Space of Square Summable Mappings/Inner Product

Definition
Let $\GF$ be a subfield of $\C$.

Let $I$ be a set.

Let $\map {\ell^2} I$ be the space of square summable mappings over $I$.

The inner product on $\map {\ell^2} I$ is the complex inner product $\innerprod \cdot \cdot: \map {\ell^2} I \times \map {\ell^2} I \to \GF$ defined by:


 * $\ds \forall f, g \in \map {\ell^2} I: \innerprod f g = \sum_{i \mathop \in I} \map f i \overline{ \map g i }$

Also see

 * Inner Product on Space of Square Summable Mappings is Complex Inner Product