example of non-separable Hilbert space
Consider real-valued functions on the real line but, instead of the usual norm, use the following inner product:
Define the Hilbert space as the set of equivalence classes of functions for which this norm is finite modulo functions for which it is zero. Note that and are orthogonal under this norm if . Hence, the set of functions , where 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 , the functions have infinite 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 sum, just as with the Fourier series of a function on a finite interval.
|Title||example of non-separable Hilbert space|
|Date of creation||2013-03-22 15:44:25|
|Last modified on||2013-03-22 15:44:25|
|Last modified by||rspuzio (6075)|