Math 202A: Lecture 9

Let V,W \in \mathrm{Ob}(\mathbf{FHil}). In this lecture we continue our proof of the fact that all Frobenius scalar products on \mathrm{Hom}(V,W) coincide:\ we fix orthonormal bases X \subset V and Y \subset W, define

\langle A,B \rangle_{XY} = \sum\limits_{x \in X} \sum\limits_{y \in Y} \overline{\langle y,Ax\rangle}\langle y,Bx \rangle, \quad A,B \in \mathrm{Hom}(V,W),

and aim to show that \langle A,B \rangle_{XY} is in fact independent of the chosen coordinate systems X and Y. We are pursuing a proof of this which is inductive in the dimension of W. In the base case, W= \mathbb{C}y, we showed in Lecture 8 that the above scalar product coincides with the Riesz scalar product on \mathrm{Hom}(V,\mathbb{C}y), which is canonically defined through vector/covector duality.

For the induction step, let V,W be fixed and also fix orthonormal bases X \subset V and Y \subset W. We know already that the corresponding Frobenius scalar product \langle \cdot,\cdot \rangle_{XY} has no dependence on Y. Fix an “anchor” vector y \in Y and let Y'=Y \backslash \{y\}. Then, a basic formula we can write down and keep in mind is

\langle A,B\rangle_{XY} = \sum\limits_{x \in X} \overline{\langle y,Ax\rangle} \langle y,Bx \rangle + \sum\limits_{x \in X} \sum\limits_{y^\prime \in Y^\prime} \overline{\langle y^\prime,Ax\rangle} \langle y^\prime,Bx \rangle.

Further note that Y^\prime is an orthonormal basis for \mathbb{C}y^\perp, the orthogonal complement of \mathbb{C}y in W. Now, for any vector w \in W we have

w= \langle y,w\rangle y + \sum\limits_{y^\prime \in Y^\prime} \langle y^\prime,w\rangle y^\prime,

where the first term is a vector in \mathbb{C}y and the remaining terms sum to a vector in \mathbb{C}y^\perp. In symbols, we have the internal direct sum decomposition

W = \mathbb{C}y \oplus \mathbb{C}y^\perp.

More generally, if H is a Hilbert space and H_1,\dots,H_r are subspaces which are pairwise orthogonal and whose union spans H, one writes

H = \bigoplus_{i=r}^r H_i.

to denote this situation. If further we have an orthonormal basis Q \subset H which decomposes as a disjoint union

Q= \bigsqcup_{i=1}^r Q_i

with Q_i a (necessarily orthonormal) basis of H_i, then we say that Q is adapted to the decomposition H=H_1 \oplus \dots \oplus H_r. In particular, the basis Y= \{y\} \sqcup Y^\prime is adapted to the orthogonal decomposition W = \mathbb{C}y \oplus \mathbb{C}y^\perp.

We want to establish that the Frobenius scalar \langle \cdot,\cdot \rangle_{XY} on \mathrm{Hom}(V,W)=\mathrm{Hom}(V,\mathbb{C}y \oplus \mathbb{C}^\perp) has no dependence on X. What the induction hypothesis actually allows us to do is the make this statement for two different spaces of linear transformations, namely \mathrm{Hom}(V,\mathbb{C}y) and \mathrm{Hom}(V,\mathbb{C}y^\perp). Namely, by induction the Frobenius scalar product \langle \cdot,\cdot\rangle_{X\{y\}} on \mathrm{Hom}(V,\mathbb{C}y) has no dependence on X, and likewise for the Frobenius scalar product \langle \cdot,\cdot\rangle_{XY^\prime} on \mathrm{Hom}(V,\mathbb{C}y^\perp). We want to glue these two statements together to make the same conclusion about \langle \cdot,\cdot \rangle_{XY} on \mathrm{Hom}(V,\mathbb{C}y \oplus \mathbb{C}^\perp).

Definition 9.1. Let H and H^\prime be Hilbert spaces. Their external direct sum H \oplus H^\prime has underlying set

H \times H^\prime = \{(h,h^\prime): h \in H,\ h^\prime \in H^\prime\},

is equipped with scalar multiplication and vector addition defined by

\alpha(h,h^\prime) = (\alpha h,\alpha h^\prime) \quad\text{and}\quad (h_1,h_1') + (h_2,h_2^\prime)=(h_1+h_2,h_1'+h_2'),

and carries the scalar product

\langle (h_1,h_1^\prime),(h_1,h_2^\prime)\rangle = \langle h_1,h_2\rangle + \langle h_1^\prime,h_2^\prime\rangle.

There are a few things here to be checked, and you should check them (for example, make sure you believe that the putative scalar product in this definition really is one).

A few basic observations are in order. First, we have an internal direct sum decomposition of the external direct sum, namely

H \oplus H^\prime = (H,0_{H^\prime}) \oplus (0_H,H^\prime),

where

(H,0_{H^\prime})=\{(h,0_{H^\prime}) \colon h \in H\}

is a subspace of H \oplus H^\prime canonically isomorphic to H, and

(0_H,H^\prime)=\{(0_H,h^\prime) \colon h^\prime \in H^\prime\}

is a subspace of H \oplus H^\prime canonically isomorphic to H^\prime. Second, let Q \subset H and Q^\prime \subset H^\prime be orthonormal bases.

Problem 9.1. Show that (Q,0_{H^\prime}) and (0_H,Q^\prime) are disjoint sets whose union constitutes an orthonormal basis of H \oplus H^\prime.

Now that we know how to glue two Hilbert spaces together additively, we can consider the Hilbert space

\mathrm{Hom}(V,\mathbb{C}y) \oplus \mathrm{Hom}(V,\mathbb{C}y^\perp),

whose scalar product

\langle \cdot,\cdot\rangle_{X\{y\}} + \langle \cdot,\cdot\rangle_{XY^\prime}

has no dependence on X because its summands have none.

Problem 9.2. Use the above to conclude that the Frobenius scalar product \langle \cdot,\cdot\rangle_{XY} on \mathrm{Hom}(V,\mathbb{C}y \oplus \mathbb{C}y^\perp) has no dependence on X.

Published by Jonathan Novak

Mathematician at UC San Diego.

Leave a Reply