example of non-separable Hilbert space

As an exaple of a Hilbert space which is not separable, one mayconsider the following function space:

Consider real-valued functions on the real line but, instead of theusual $L^2$ norm, use the following inner product:

$$(f,g)=\underset{R\to \mathrm{\infty }}{lim}\frac{1}{R}{\int }_{-R}^{+R}f(x)g(x)𝑑x$$

The first thing to note about this is that non-trivial functions havenorm 0. For instance, any function of $L^2$ has zero norm accordingto this inner product.

Define the Hilbert space as the set of equivalence classes offunctions for which this norm is finite modulo functions for which itis zero. Note that $\sin ax$ and $\sin bx$ are orthogonal under thisnorm if $a\ne b$. Hence, the set of functions $\sin ax$, where $a$is a real number, form an orthonormal set. Since the number of realnumbers is uncountable, we have an uncountably infinite orthonormal set,so this Hilbert space is not separable.

It is important not to confuse what we are doing here with the Fourierintegral. In that case, we are dealing with $L^2$, the functions$\sin ax$ have infinite $L^2$ norm (so they are not elements of thatHilbert space) and the expansion of a function in terms of them is adirect integral. By contrast, in the case propounded here, theexpansion of a function of this space in terms of them would take theform of a direct sum, just as with the Fourier series of a function ona finite interval.