M346 Videos -- Section 6.8

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Videos for Section 6.8

This section features one video on the topic of Hilbert Spaces and Hilbert bases. A Hilbert space is a vector space (usually infinite dimensional) with an inner product, and with a topology that comes from that inner product. (Technically, it is a complete normed vector space, a.k.a. Banach space, with the norm coming from an inner product.) By "topology" I mean that we have a notion on limits and convergence.

If $\{{\bf x_1, x_2, \ldots}\}$ is a sequence of vectors in a Hilbert space $\mathcal{H}$ , we say that the sequence converges to a vector ${\bf y}$ if (and only if) $|{\bf y - x_i}| \to 0$ , or equivalently if

$\langle {\bf y - x_i | y - x_i} \rangle \to 0.$ Once we understand convergence, we can talk about infinite sums and infinite linear combinations. As with numbers, an infinite sum

$\sum_{n=1}^\infty c_n {\bf b}_n$ of vectors is the limit of a finite sum

$\sum_{n=1}^\infty c_n {\bf b}_n = \lim_{N \to \infty} \sum_{n=1}^N c_n {\bf b}_n,$ as long as the sum converges. The sum converges absolutely if

$\sum |c_n| |{\bf b}_n|$ converges, and there's a theorem (that we won't prove) that absolute convergence implies convergence.

Now that we're allowed to take infinite linear combinations, we can use them to expand vectors using an infinite basis. A Hilbert basis is a set $\{{\bf b_1, b_2, \ldots}\}$ of vectors in our Hilbert space $\mathcal{H}$ such that

Every vector in $\mathcal{H}$ can be written as a linear combination $\sum_{n=1}^\infty c_n {\bf b_n}$ , and
The only way to write $0 = \sum c_n {\bf b_n}$ is if all of the coefficients $c_n$ are zero.

These are the analogs of the usual notions of spanning and being linearly independent, only with infinite sums instead of finite sums. The following video lays out the theory of Hilbert spaces, and explores three examples:

The three examples are:

$\ell_2$ is the space of infinite sequences ${\bf x} = (x_1, x_2, x_3, \ldots, )$ of complex numbers such that $\sum |x_n|^2$ converges. The inner product is $\langle {\bf x} | {\bf y} \rangle = \sum_{j=1}^\infty \bar x_j y_j$ . This is like the standard inner product on $\mathbb{C}^n$ , only with infinite sums instead of finite sums. A Hilbert basis consists of ${\bf e_1} = (1,0,0,\ldots)$ , ${\bf e_2} = (0,1,0,0,\ldots)$ , etc.
$L^2(\mathbb{R})$ is the space of all reasonable functions $f(t)$ on the real line such that $\int_{-\infty}^\infty |f(t)|^2 dt$ converges. (We'll ignore the technical questions of what sorts of functions qualify as reasonable. See the book for a more careful treatment.) The inner product is $\langle {\bf f} | {\bf g} \rangle = \int_{-\infty}^\infty \overline{f(t)} g(t) dt.$ It turns out that the functions $t^n e^{-t^2/2}$ , where $n$ ranges from 0 to $\infty$ , form a Hilbert basis. (This is not obvious, and is related to finding the energy levels of a harmonic oscillator in quantum mechanics.)
$L^2([0,L])$ is like $L^2(\mathbb{R})$ , except that we only consider functions on the interval $[0,L]$ and do our integrals from $t=0$ to $t=L$ . It turns out that the functions $\xi_n(t) = \sin(n \pi t/L)$ form an orthogonal Hilbert basis for this space. Expanding vectors with respect to this basis is called Fourier Series, and is the topic of our next section.