Notes on Chapter 4 (Elementary Hilbert Space Theory) of Walter Rudin's Real and Complex Analysis. I'm also gonna try to style these a bit more nicely!

Chapter 4

An inner product space that is complete w.r.t to the metric arising from the inner product is a Hilbert space (\(H\) will always be a Hilbert space in this post unless otherwise stated). Various basic properties of Hilbert spaces and convex sets are explained. We have the following:

Minimal element in convex set: Every nonempty, closed, convex set in a Hilbert space contains a unique element of smallest norm. □

Projections: A closed subspace \(M\) of a Hilbert space gives rise to projections on to it and its orthogonal complement \(M^\bot\). The complements span the whole space. □

Representing linear functionals: If \(L\) is a continuous linear functional on \(H\), then there is a unique \(y\in H\) s.t. \(Lx = (x, y)\) forall \(x\in H\). □

If \(A\ni\alpha\) indexes an orthonormal set \(\{u_\alpha\}\), denote by \(\hat{x}(\alpha)\) the inner product \((x, u_\alpha)\), i.e. the coordinates of \(x\). Following this, we can obtain the following theorems about orthonormal sets:

Bessel Inequality, Riesz-Fischer: Let \(\{u_\alpha: \alpha\in A\}\) be an orthonormal set in \(H\), and let \(P\) be the space of all finite linear combinations of the vectors \(\mu_\alpha\). The inequality $$\sum_{\alpha\in A} |\hat{x}(\alpha)| \leq \|x\|^2\qquad\text{\small(Bessel)}$$ holds then for every \(x\in H\), and \(x\rightarrow\hat{x}\) is a continuous linear mapping of \(H\) onto \(\ell^2(A)\) whose restriction to the closure \(\bar{P}\) of \(P\) is an isometry of \(\bar{P}\) onto \(\ell^2(A)\). □

Using that, we can derive the following specialization:

Equivalent conditions for orthonormal basis: Let \(\{u_\alpha: \alpha\in A\}\) be an orthonormal set in \(H\). Each of the following are equivalent:

• \(\{u_\alpha\}\) is a maximal orthonormal set in \(H\).
• The set \(P\) of all finite linear combinations of members of \(\{u_\alpha\}\) is dense in \(H\).
• The equality $$\sum_{\alpha\in A} |\hat{x}(\alpha)|^2 = \|x\|^2$$ holds for all \(x\in H\).
• The equality $$\sum_{\alpha\in A} \hat{x}(\alpha)\overline{\hat{y}(\alpha)} = (x, y)\qquad\text{\small(Parseval)}$$ holds for all \(x, y\in H\). □

Finally, with the help of Zorn's Lemma, we get existence:

Orthonormal base existence: Every Hilbert space has an orthonormal basis, in particular every orthonormal set can be extended to an orthonormal basis. □

Together these show that Hilbert spaces (even infinite-dimensional ones) basically have the nice properties you would expect of something like \(\mathbb{R}^n\).

Finally, we develop the basic theory of Fourier series in \(L^2(T)\) (\(T\) being the unit circle in the complex plane), so equivalently, \(2\pi\)-periodic functions on \(\mathbb{R}\). This turns out to fall out nicely from our previous discussion on Hilbert spaces. First, we note that the \(u_n(t)=e^{int}\) with the inner product on \(L^2(T)\) defined as $$(f,g)=\frac{1}{2\pi}\int_{-\pi}^\pi f(t)\overline{g(t)}\ dt$$ form an orthonormal set in \(L^2(T)\), called the trigonometric system.

By Riesz-Fischer, if we show that the set of all trigonometric polynomials is dense in \(L^2(T)\), then it follows that the \(u_n\) form a basis for \(L^2(T)\). By the Approximation by \(C_c(X)\) theorem, \(C(T)\) is dense in \(L^2(T)\), so it suffices to show that there is a trigonometric polynomial \(P\) s.t. \(\|f-P\|_2<\varepsilon\) for \(f\in C(T)\), which will follow from \(\|f-P\|_\infty<\varepsilon\):

Approximation by Trigonometric Polynomials: If \(f\in C(T)\) and \(\varepsilon>0\), there is a trigonometric polynomial \(P\) s.t. \(|f(t)-P(t)|<\varepsilon\) for all real \(t\). □

This can essentially be done by constructing a series of trigonometric 'bump' functions that approximate the Dirac delta in the limit, and then taking the convolution of that with \(f\), giving a series of trigonometric polynomials that tend to \(f\). We can now specialize what we know to the usual Fourier series.

For any \(f\in L^1(T)\), the Fourier coefficients are

$$\hat{f}(n)=\frac{1}{2\pi}\int_{-\pi}^\pi f(t)e^{-int}\ dt,$$

The Fourier series of \(f\) is then

$$\sum_{-\infty}^\infty \hat{f}(n)e^{int},$$

with the partial sums being

$$s_N(t)=\sum_{-N}^N \hat{f}(n)e^{int}.$$

Given this, we have the following 'concrete' theorems:

Riesz-Fischer: If \{c_n\} is a sequence of \(L^2\)-summable complex numbers, then there exists an \(f\in L^2(T)\) s.t. $$c_n=\frac{1}{2\pi}\int_{-\pi}^\pi f(t)e^{-int}\ dt\qquad(n\in Z).$$ □

Parseval: We also have $$\sum_{n=-\infty}^\infty \hat{f}(n)\overline{\hat{g}(n)}=\frac{1}{2\pi}\int_{-\pi}^\pi f(t)\overline{g(t)}\ dt$$ and $$\lim_{N\rightarrow\infty} \|f-s_N\|_2=0.$$ □

To summarize, \(f\leftrightarrow\hat{f}\) is a Hilbert space isomorphism of \(L^2(T)\) onto \(\ell^2(Z)\). Thanks to the completeness of \(L^2\), we were able to get these results.