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1. We used the following result in our class discussion with a promise that you'd derive it in the homework. Let V be a Hilbert space with subspace W. Suppose U : W → V is a linear operator that preserves inner products. Show that there exists a unitary operator U' : V → V with the property that U'lw) = U|w) for |w) e W but with U' defined on all of V.

1. We used the following result in our class discussion with a promise that you'd derive it in the homework. Let V be a Hilbert space with subspace W. Suppose U : W → V is a linear operator that preserves inner products. Show that there exists a unitary operator U' : V → V with the property that U'lw) = U|w) for |w) e W but with U' defined on all of V.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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1. We used the following result in our class discussion with a promise that you'd derive it in the homework. Let V be a Hilbert space with subspace W. Suppose U : W → V is a linear operator that preserves inner products. Show that there exists a unitary operator U' : V → V with the property that U'w) = U|w) for |w) E W but with U' defined on all of V.
Transcribed Image Text:1. We used the following result in our class discussion with a promise that you'd derive it in the homework. Let V be a Hilbert space with subspace W. Suppose U : W → V is a linear operator that preserves inner products. Show that there exists a unitary operator U' : V → V with the property that U'w) = U|w) for |w) E W but with U' defined on all of V.
Expert Solution
Step 1

Given that, V be a Hilbert space with subspace W.

So, U is defined on a subspace W and preserves inner product, we have  Uw=w∀w∈W

Since, U is invertible then U is uniformly continuous,

That is U is a homeomorphism from W onto UW.

Now by uniform continuity U extends to a linear operator on closure of W.

Since inner product is jointly continuous, this extension also preserves  the inner product.

Therefore without loss of generality, we assume  that W is closed and U is a linear operator on W which preserves inner product.

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