Two Prototypical Examples

The two most important Hilbert spaces are:

1. The vector space of square summable sequences
   $$\ell^2 =\{ U=(u_1,u_2,\dots )\colon u_i\in\{\,\textrm{complex~numbers}\,\},~ i=1,2,\dots\}$$
   with squared norm given by the inner product
   $$\Vert U\Vert^2 = \langle U,U\rangle =\sum^\infty_{i=1}\vert u_i\vert^2\,.$$
   This space, which is important in signal processing, among others, is discussed in Dettman.
2. The vector space of square integrable functions
   $$L^2(a,b)=\{ f\colon \int^b_a\overline{f}f\rho (x)dx<\infty\,;~\rho (x)>0\}.$$
   The positive function $\rho (x)$ is given. It is called a weight function.

   This vector space has the following three properties.

   1. $L^2(a,b)$ is an inner product space
      $$\begin{array}{rclcl} \langle f,g\rangle &\equiv &\int^b_a\overlin... ... (x)\,dx &&\textrm{which~ is~the~squared}\\ &&&&\textit{norm~of~}f \end{array}$$
      Comment: $\rho (x)>0\Rightarrow\sqrt{\rho}$ can be absorbed into the functions, so that instead of $\{ f(x)\}$ one has $\{ h(x)\}=\{ f(x)\sqrt{\rho (x)}\}$ with the squared norm
      $$\langle h,h\rangle = \int^b_a\vert h\vert^2\,dx.$$

      Conclusion: We still have the same inner product space.

   2. $L^2(a,b)$ is closed under addition:
      1. Expanding the inner product of a sum with itself, we have
         

         $$\Vert f+g\Vert^2=\langle f+g,f+g\rangle = \Vert f\Vert^2+\Vert g\Vert^2+ \langle f,g\rangle +\langle g,f\rangle$$

         $$= \Vert f\Vert^2+\Vert g\Vert^2 +2\textrm{Re}\,\langle f,g\rangle$$

         $$\le \Vert f\Vert^2+\Vert g\Vert^2 +2\Vert f\Vert\,\Vert g\Vert ,$$ (15)

         
         where we used the Cauchy-Schwarz inequality.
      2. Recall that
         

         $$\le (\Vert f\Vert -\Vert g\Vert )^2=\Vert f\Vert^2+\Vert g\Vert^2-2\Vert f\Vert\,\Vert g\Vert .$$ 0 (16)

         
         
      3. Eqs.(3.20) and (3.21) $\Rightarrow\Vert f+g\Vert^2\le 2\Vert f\Vert^2 +2\Vert g\Vert^2 .$
         Thus we have
         

         $$~ f,g~~\textrm{square~integrable} \Rightarrow f+g~~\textrm{is square~integrable}\, ,$$

         
         i.e. $L^2$ is indeed closed under addition.

   3. $L^2(a,b)$ is Cauchy complete.