Let be Hilbert spaces with inner products and , respectively, and corresponding norms and . A linear transformation is called unitary if
- is bijective.
- for all .
is bijective, and the second property shows that , hence is unitary.
Using the polarization identity for inner product spaces, we obtain the following equivalence for the second property.
Lemma 1. is a unitary mapping if and only if is a bijective linear transformation satisfying
We say that two Hilbert spaces are unitarily equivalent or unitarily isomorphic if there exists a unitary mapping .
We now turn to weighted spaces, which will talk more about when we deal with Sturm-Liouville problems. Let be a strictly positive continuous function. Define the to be the space of all measurable functions such that
I claim that we can define an inner product on by
The linearity in the first argument and the conjugate linearity in the second argument follows from the observation that , where denotes the standard inner product. That follows from the nonnegativity of . Equality holds in the preceding inequality if and only if since for all by Weierstrass’ extreme value theorem.
To see that is complete and therefore a Hilbert space, let be a Cauchy sequence with respect to the norm . It suffices to show that has a subsequence which converges to a function . We can choose a strictly increasing sequence of indices such that
Note that by Minkowski’s inequality,
By the monotone converge theorem, converges a.e. and is in . By the dominated convergence theorem,
Set . It is clear that and
We summarize our results in the following theorem.
Theorem 2. For a fixed strictly positive, continuous function , is a Hilbert space.
Above, we restricted ourselves to the case where was strictly positive and continuous, but we can relax this hypothesis to is a strictly positive and measurable on . The only nontrivial verification is that implies that a.e. Recall that for a nonnegative measurable function ,
Applying this result to , we see that
where the penultimate equivalence follows from the strict positivity of .
It might not surprise the reader that and are unitarily equvalent. Define a linear transformation
It is clear that is a linear transformation and
from which injectivity also follows. To see that is surjective, for , define . We just need to verify that . Indeed,
We summarize this result with our last theorem.
Theorem 3. For a fixed strictly positive, measurable function , and are unitarily equivalent.