Hilbert Space formalism

Summary#

All periodic functions on \(\mathbb{R}\), up to a scaling, can be seen as functions on \(\mathbb{T} = \mathbb{R}/\mathbb{Z}\). The space \(C(\mathbb{T})\) equipped with the inner product \(\langle\cdot,\cdot\rangle\) defined by 

\[\langle f,g\rangle = \int_0^1 f(x)\overline{g(x)} dx\]

is a pre-Hilbert space (with the norm, called \(L^2\)-norm, defined by \(||x|| = \sqrt{\langle x,x\rangle}\)). Upon completion with respect to the norm it becomes the Hilbert space \(L^2(\mathbb{T})\). The family of functions \(e_k(x) := e^{2\pi i k x}\) forms an orthonormal basis of the Hilbert space \(L^2(\mathbb{T})\).

\(L^2\)-convergence#

Recall that the sequence \(f_n\) converges in the \(L^2\)-norm to \(f\) if 

\[\lim_{n\to\infty}||f - f_n||_2 = 0\]

 where \(||\cdot||_2\) is the \(L^2\)-norm. For \(L^2\)-norm to make sense we require \(f\) and \(f_n\) to be Lebesgue integrable. We also remark the following simple but useful result:

Theorem. If \(f_n \to f\) uniformly on \([a,b]\), then \(f_n \to f\) in norm.

We have used \(PC([a,b])\) before, but as a normed space it is not complete. The appropriate space to use is the space \(L^2([a,b])\) of square Lebesgue integrable functions.

Theorem. \(L^2([a,b])\) is complete w.r.t. convergence in \(L^2\)-norm. For any \(f\in L^2([a,b])\), there is a sequence \(f_n\) of continuous functions on \([a,b]\) s.t. \(f_n \to f\) in norm; actually, these can be the restriction to \([a,b]\) of smooth functions.