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## Isomorphic Hilbert Spaces

Lecture 9

The completeness relation, Eq. (1.17), is a remarkable result! It implies the generalized Fourier expansion

with . Indeed, the completeness relation implies

Subtracting and adding the limit of the sum

one has
 0 0 0

which is what is meant by

But there is more. The geometrical significance of this generalized Fourier expansion is astonishing. It is a one-to-one linear correspondence - let us call it - between the Hilbert space of square integrable functions on and the Hilbert space of square summable series (functions on the integers''). The correspondence

(i) is one-to-one and onto, which means it has an inverse:

(ii) is linear, which means it takes closed triangles in into closed triangles in :

(iii) preserves lengths. Indeed,

implies

It follows that

Consequently, preserves lengths, inner products, and angles (if the Hilbert space is real).

Definition: A linear transformation which is one-to-one and onto is called an isomorphism.

Definition: A distance preserving transformation between two metric spaces is called an isometric transformation, or simply an isometry.

In that case, the two spaces are said to be isometric spaces. This means they look the same from the viewpoint of geometry.

To summarize, the striking feature of the completeness, i.e., Parseval's relation is that it establishes an isometric isomorphism, or more briefly an isometry between the two spaces.

Thus

They are geometrically the same (right triangles in one space correspond to right triangles in the other space).

Because one can establish a linear isometry between any Hilbert space and one and the same , the space of square summable series, one obtains the

Theorem: (Isomorphism theorem) Any two complex Hilbert spaces are isomorphic. In fact, so are any two real Hilbert spaces.

Lecture 10

Comment: The isometric isomorphism is a unitary transformation whose elements are . Indeed, consider the equation

The coefficients are the components of an infinite dimensional column vector in . The function is an infinite dimensional column vector whose components are labelled by the continuous index . It follows that are the entries of a matrix whose columns are labelled by and whose rows are labelled by . The orthogonality conditions

expresses the orthonormality of the colums of this matrix. The completeness relation

expresses the orthonormality of the rows. It follows that represents a unitary transformation which maps onto .

Exercise 15.1 (SQUARED LENGTHS AND INNER PRODUCTS)
An isometry between the Hilbert space of square integrable functions , and the Hilbert space of square summable sequences is a linear one-to-one and onto transformation with the property that it preserves squared lengths:

SHOW that

where

Exercise 15.2
Let be a fixed and given square integrable function, i.e.

One can think of as a function whose non-zero values are concentrated in a small set around the origin .

Consider the concomitant windowed'' Fourier transform on , the space of square integrable functions,

Let be an element of the range space . It is evident that

is an inner product on .

FIND a formula for in terms of the inner product

on .

Exercise 15.3 (SHANNON'S SAMPLING FUNCTIONS)
By

(a)
starting with the orthonormality relation

where

(b)
then rescaling the integration domain by introducing the variable

where is a fixed constant (the "band width"),

(c)
and finally going to the limits .

(i)
Show that the set of functions

is an orthogonal set satisfying

What is ?

(ii)
This set of functions

is not complete on , but it is complete on a certain subset .

What is this subset? i.e. What property must a function have in order that ?

This question can be answered with the help of Parseval's (completeness'') relation as follows: Recall that completeness on here means that implies that one can write as

with , which we know is equivalent to

 (119)

Thus, to answer the question, we must ask and answer: What property must have in order to guarantee that Eq.(1.19) be satisfied? Therefore, to give a correct answer, one must (i) identify the property and (ii) then show that Parseval's relation is satisfied by every such function.

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Next: Fourier Theory Up: Hilbert Spaces Previous: Recapitulation   Contents   Index
Ulrich Gerlach 2010-12-09