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.