Math 202A: Review Lecture 2

Let \mathbf{FHil} be the category whose objects are complex vector spaces V equipped with a scalar product \langle \cdot,\cdot \rangle and whose morphisms are linear transformations. We call pairs (V,\langle \cdot,\cdot \rangle) Hilbert spaces, and our convention is that the scalar product is linear in the second slot. The standard definition of Hilbert space includes an extra clause which we do not need and will omit (see below),

Problem 2.1. Prove that A \in \mathrm{Hom}(V,W) is an isomorphism if and only if it is a bijection. That is, isomorphisms in \mathbf{FHil} are precisely linear bijections.

The Hilbert space norm on V is defined by

\|v\| = \sqrt{\langle v,v \rangle}.

To prove that this really is a norm, one needs the Cauchy-Schwarz inequality in order to verify the triangle inequality,

\|v+w\|^2 = \|v\|^2 + 2\Re \langle v,w\rangle + \|w\|^2 \leq \|v\|^2 + 2\|v\|\|w\| + \|w\|^2 = (\|v\|+\|w\|)^2.

Problem 2.2. Prove the reverse triangle inequality: \left| \|v\|-\|w\| \right|\leq \|v-w\|. Make sure you understand why this implies that the function V \to \mathbb{C} defined by v \mapsto \|v\| is continuous, in fact 1-Lipschitz.

An important fact is that the scalar product in a Hilbert space V can be recovered from the norm.

Theorem 2.3. (Polarization) For any v,w \in V, we have

\langle v,w \rangle = \frac{1}{4}\left( \|v+w\|^2 - \|v-w\|^2 -i\|v+iw\|^2+i\|v-iw\|^2 \right).

Two vectors v,w in Hilbert space V are said to be orthogonal if \langle v,w \rangle=0.

Theorem 2.4. (Pythagoras) For any orthogonal vectors v,w \in V, we have \|v+w\|^2=\|v\|^2 + \|w\|^2.

An important generalization of the Pythagorean theorem is the following.

Theorem 2.5 (Parallelogram Law) For any v,w \in V, we have \|v+w\|^2+\|v-w\|^2=2\|v\|^2+2\|w\|^2.

Much of the importance of the Parallelogram Law is that it characterizes when a normed vector space is a Hilbert space. This remarkable fact was first noticed by von Neumann.

Problem 2.3. If V is a complex vector space equipped with a norm \|\cdot\|, then there exists a scalar product on V such that \|v\|^2=\langle v,v \rangle if and only if the Parallelogram Law holds.

The standard definition of Hilbert space includes an extra completeness condition which we was not used above, and will not be used going forward. Therefore, we omit this condition from our definition of what constitutes a Hilbert space.

We define the quantization functor

\mathcal{F} \colon \mathbf{FSet} \longrightarrow \mathbf{Hil}

as follows. First, for every finite set X we declare

\mathcal{F}(X) = \{a \colon X \to \mathbb{C}\}

to be the vector space of complex-valued functions on X together with the scalar product

\langle a,b\rangle = \sum\limits_{x \in X} \overline{a(x)}b(x).

Second, if X,Y are finite sets and f \in \mathrm{Hom}(X,Y) is a function, then we declare \mathcal{F}(f) \in \mathrm{Hom}(\mathcal{F}(X),\mathcal{F}(Y)) to be the linear transformation defined by

[\mathcal{F}(f)a](y) = \sum\limits_{x \in f^{-1}(y)} a(x).

In words, [\mathcal{F}(f)a](y) is the sum of the values of a \in \mathcal{F}(X) over the points in the fiber of f over y. Assuming X,Y are disjoint, you can visualize this by associating to f \in \mathrm{Hom}(X,Y) the bipartite graph with vertex set X \sqcup Y and edges \{x,f(x)\} given by input-output pairs of f. This bipartite graph is a disjoint union of star graphs: each star consists of a hub vertex y \in Y, and the remaining vertices of the star are the points of X which map to y via f. The function \mathcal{F}(f)a acts on Y by summing the values of a over non-hub vertices.

Problem 2.4. Prove that finite sets X and Y are isomorphic if and only if the Hilbert spaces \mathcal{F}(X) and \mathcal{F}(Y) are isomorphic.

Now let us explain why we refer to \mathcal{F} as the quantization functor. For each x \in X, the corresponding elementary function e_x \in \mathcal{F}(X) is defined by

e_x(x^\prime) = \delta_{xx^\prime}.

A basic feature of these functions is their orthogonality,

\langle e_{x_1},e_{x_2} \rangle = \delta_{x_1x_2}.

Theorem 2.6. For any a \in \mathcal{F}(X), there exist unique scalars \alpha_x, x \in X such that

a = \sum\limits_{x \in X} \alpha_xe_x.

Proof: To prove existence, check that \alpha_x=a(x) works. For uniqueness, imagine we have two such representations of a and then use evaluations of a to show these representations are the same. \square

Theorem 2.6 allow us to develop an alternative perspective which is widely used in algebra: we view complex-valued functions on X as formal \mathbb{C}-linear combinations of its points,

a=\sum\limits_{x \in X} \alpha_x x.

This is really just notation, and one should remember that the expression above is just a different way of writing down the function a(x)=\alpha_x, x \in X. The formal linear combination perspective on functions is also widely used in physics. A set X, for example the natural set X=[n], can be viewed as the state space of a particle \bullet on the integer lattice which may be located at any of the sites 1,\dots,n. The particle \bullet is “classical” in the sense of classical mechanics: it has a definite location, which may be any of the sites 1,\dots,n. A quantum particle does not have a definition location: before it is observed, it exists simultaneously in each of the states 1,\dots,n,, and only after it is observed does its location become definite. A (pure) quantum state is a formal linear combination

a = \sum\limits_{x \in X} \alpha_x x

of classical states x such that

\sum\limits_{x \in X} |\alpha_x|^2 =1,

where the amplitude |\alpha_x|^2 is the probability that \bullet will be in state x once it is observed. Thus, (pure) quantum states are the same thing as unit vectors in the Hilbert space \mathcal{F}(X). In applying the functor \mathcal{F}, we are passing from the state space X of a classical particle to the state space \mathcal{F}(X) of a quantum particle.

Mathematically, many features of Hilbert space can be viewed as quantum generalizations of familiar set-theoretic properties. For example, let X_1 and X_2 be two subsets of X, and let

a_1 = \sum\limits_{x \in X_1} x \text{ and } a_2=\sum\limits_{x \in X_2} x

be their indicator functions in \mathcal{F}(X). whose norms are \|a_1\|^2=|X_1| and \|a_2\|^2=|X_2|. If X_1 and X_2 are disjoint, then a_1 and a_2 are orthogonal in \mathcal{F}(X), and the Pythogorean theorem

\|a_1 + a_2\|^2 = \|a_1\|^2 + \|a_2\|^2

reproduces |X \sqcup Y|=|X|+|Y|. However, the Pythagorean theorem holds for all pairs of orthogonal vectors in \mathcal{F}(X), not just indicator functions of disjoint subsets of X. Similarly, the Parallelogram Law can be seen as a quantum generalization of the inclusion-exclusion principle.

Definition 2.7. A Hilbert space V is said to be finite-dimensional if there exists a finite set X such that V is isomorphic to \mathcal{F}(X). Otherwise, it is said to be infinite-dimensional.

This definition is non-traditional, but it has certain advantages. In particular, thanks to Problem 2.4 we can define the dimension of a finite-dimensional Hilbert space V to be the cardinality of any finite set X such that V is isomorphic to \mathcal{F}(X).

Problem 2.4. Prove that finite-dimensional Hilbert spaces V and W are isomorphic if and only if they have the same dimension.

A more traditional approach to dimension is based on the following definition.

Definition 2.8. A finite set S of vectors in a Hilbert space V is said to be linearly independent if

\sum\limits_{s \in S} \alpha_s s=0_V \implies \alpha_x=0 \text{ for all }s \in S.

It is easy to see that any finite orthonormal set X \subset V is linear independent. The converse is also true.

Proposition 2.9. (Gram-Schmidt) If V contains a linearly independent set of cardinality n, then it contains an orthonormal set of cardinality n.

From here we establish the equivalence of Definition 2.7 and the standard definition of dimension.

Theorem 2.10. A Hilbert space V has dimension n \in \mathbb{N} if and only if it contains an orthonormal set of cardinality n and does not contain an orthonormal set of cardinality n+1.

Published by Jonathan Novak

Mathematician at UC San Diego.

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