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 $f: I \to \GF$.

Inner Product

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

• Results about Space of Square Summable Mappings can be found here.
• 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

Sources

• 1990: John B. Conway: A Course in Functional Analysis (2nd ed.) ... (previous) ... (next): $\text{I}$ Hilbert Spaces: $\S 1.$ Elementary Properties and Examples: Example $1.7$