\(\ell_2\) Inner Product Space

To get an Inner Product Space, we need our vectors to have a finite length, restrict attention to sequences for which \(\sum_i |a^i|^2 <\infty\) and define \(\langle a|b\rangle = \sum_i a^{i*}b^i\). 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 \(\mathbb{C}^\infty\). This is our first example-and the prototype-of a Hilbert Space.