example of non-separable Hilbert space

As an exaple of a Hilbert spaceMathworldPlanetmath which is not separablePlanetmathPlanetmath, one may consider the following function space:

Consider real-valued functions on the real line but, instead of the usual L2 norm, use the following inner productMathworldPlanetmath:


The first thing to note about this is that non-trivial functions have norm 0. For instance, any function of L2 has zero norm according to this inner product.

Define the Hilbert space as the set of equivalence classesMathworldPlanetmathPlanetmath of functions for which this norm is finite modulo functions for which it is zero. Note that sinax and sinbx are orthogonalMathworldPlanetmathPlanetmath under this norm if ab. Hence, the set of functions sinax, where a is a real number, form an orthonormal set. Since the number of real numbers 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 Fourier integral. In that case, we are dealing with L2, the functions sinax have infiniteMathworldPlanetmathPlanetmath L2 norm (so they are not elements of that Hilbert space) and the expansion of a function in terms of them is a direct integral. By contrast, in the case propounded here, the expansion of a function of this space in terms of them would take the form of a direct sumMathworldPlanetmath, just as with the Fourier series of a function on a finite interval.

Title example of non-separable Hilbert space
Canonical name ExampleOfNonseparableHilbertSpace
Date of creation 2013-03-22 15:44:25
Last modified on 2013-03-22 15:44:25
Owner rspuzio (6075)
Last modified by rspuzio (6075)
Numerical id 6
Author rspuzio (6075)
Entry type Example
Classification msc 46C05
Related topic AlmostPeriodicFunction