${\ell }_2$ Inner Product Space

To get an Inner Product Space, we need our vectors to have a finite length, restrict attention to sequences for which $\sum _i|a^i{|}^2<\mathrm{\infty }$ and define $⟨a|b⟩=\sum _i{a}^{i\ast }{b}^i$. This is guaranteed to be well defined by $\sum _i|a^i{|}^2$ is finite (homework).

${\ell }_p\equiv \{\{x_i\}:x_i\in \mathbb{C},\sum _i|x_i{|}^p\text{ finite}\}$. It is a proper subspace of ${\mathbb{C}}^{\mathrm{\infty }}$. This is our first example-and the prototype-of a Hilbert Space.