Definition:Space of Square Summable Mappings

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

Let $I$ be a set.

The space of square summable mappings over $I$, denoted $\map {\ell^2} I$, is the set:


 * $\ds \map {\ell^2} I := \set{ f: I \to \GF: \sum_{i \mathop \in I} \cmod{ \map f i }^2 < \infty }$

of square summable mappings on $I$, considered as a vector subspace of the vector space $\GF^I$ of all mappings $f: I \to \GF$.

Also see

 * Definition:$p$-Sequence Space


 * Space of Square Summable Mappings is Vector Space
 * Space of Square Summable Mappings is $L^2$ Space
 * Space of Square Summable Mappings is Hilbert Space