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The backward direction is easy. As for the forward direction, I don't understand how given an arbitrary vector space, you can go about defining an inner product without knowing something more about it.

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- Thread starter Treadstone 71
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- #1

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The backward direction is easy. As for the forward direction, I don't understand how given an arbitrary vector space, you can go about defining an inner product without knowing something more about it.

- #2

AKG

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Now the problem is, how to find an inner product such that T is self-adjoint relative to it. Well what things do you know about T, given that it's diagonalizable? Second, you're not going to write out what <v,w> is for each individual v and w in V. Given that inner products are multilinear, it suffices to define an inner product on a ______. But you should know that if T is diagonalizable, there is a ______ with some strong relation to T. Fill in the blank, figure out what that "strong relation" is, and use it to prove that T is self-adjoint w.r.t. your inner product.

- #3

Hurkyl

Staff Emeritus

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TheAs for the forward direction, I don't understand how given an arbitrary vector space, you can go about defining an inner product without knowing something more about it.

Anyways, I think the backward direction gives you a

- #4

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Got it. Thanks.

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