Space of Square Summable Mappings is Hilbert Space

Theorem

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 $I$.

Let $\innerprod \cdot \cdot: \map {\ell^2} I \times \map {\ell^2} I \to \GF$ be the inner product on $\map {\ell^2} I$.

Then $\map {\ell^2} I$ endowed with $\innerprod \cdot \cdot$ is a Hilbert space.

Proof

By Space of Square Summable Mappings is $L^2$ Space, $\map {\ell^2} I$ is equal to $\map {L^2} {I, \powerset I, \mu}$.

The result follows by $L^2$ Space forms Hilbert Space.

$\blacksquare$

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 $1.7$