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My question is in reference to the derivation below from the physics site. It shows how the metric tensor raises and lowers indices. I cut it off halfway through because I didn't think the full derivation was relevant to the question, but here it is, just in case. I don't believe my question is a duplicate.

enter image description here

My question is, why is the map $\tau$ considered to be "natural"? Does the fact that the vector space has an inner product necessitate that the dual has one? (maybe so that there's an isomorphism?)

Edit: I'm asking if the fact that the vector space has an inner product induces the need for its dual to have an inner product to create an isomorphism.

Shocked
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  • I don't think I'm asking the same thing, here. Forgive me if I am. – Shocked Aug 27 '18 at 16:00
  • If $V$ is finite dimensional, then $\tau$ is indeed an isomorphism between $V$ and $V^$. Then you can easily define an inner product on $V^$ via $\langle \varphi, \psi \rangle := (\tau^{-1}(\varphi),\tau^{-1}(\psi))$. But that's not really interesting, is it? – Babelfish Aug 27 '18 at 16:06
  • Does natural mean something to you? There are sophisticated definitions for naturality, but if you are only concerned with linear algebra, then "does not depend on some choice of a basis or something similar" is the main interpretation of "natural". – Babelfish Aug 27 '18 at 16:07
  • OK this is where my confusion came in. I thought that $\tau$ was the inner product that was defined on the dual. First, where does $\tau$ live, if not in the dual space. Also, what makes it natural? Edit: I think I got it. – Shocked Aug 27 '18 at 16:09
  • $\tau$ is the inner product, where you fix the first coordinate. It is a linear map from $V$ to $V^$: $\tau \colon V\to V^$, $\tau(x) \in V^*$, where $\tau(x)(v) = (x,v)$. \\\ It is natural since it is induced by the inner product. – Babelfish Aug 27 '18 at 16:11
  • @Babelfish most recent comment: Thanks. That's the connection I was looking for. I just wanted confirmation for what I was thinking. Thank you. – Shocked Aug 27 '18 at 16:13
  • Let us continue this discussion in chat. – Shocked Aug 27 '18 at 16:18
  • @LordSharktheUnknown Not Accepted. This question has further relavance. – Nick Aug 27 '18 at 20:47

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