Basis (Hilbert Space)/Examples/Space of Square Summable Mappings

Example of Basis (Hilbert Space)

Let $I$ be a set.

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

For $i \in I$, define $e_i: I \to \GF$ as:

$\map {e_i} j := \begin{cases} 1 &: i = j \\ 0 &: i \ne j \end{cases}$

Then $\set{ e_i : i \in I}$ is a basis for $\map {\ell^2} I$.

Proof

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Sources

• 1990: John B. Conway: A Course in Functional Analysis (2nd ed.) ... (previous) ... (next): $\text{I}$ Hilbert Spaces: $\S 4.$ Orthonormal Sets of Vectors and Bases: Example $4.5$