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title
Every Single Problem
description
Math musings
site name
Every Single Problem
author
updated
2026-03-04 04:14:32
raw text
Every Single Problem – Math musings Skip to content Every Single Problem Math musings Why are inner products symmetric? So when I got to grad school, certain gaps in my (very unconventional) math education revealed themselves, and one of them was that I didn’t know anything about vector space duality. I took a graduate linear algebra course that first semester, but I think what actually caught me up to speed was my peers. I have a vivid memory of a conversation with Alex Blumenthal in which he said something to the effect of: “You know what an inner product really is though?? It’s an isomorphism to the dual!!! “ So, that was dope. Thanks, Alex! But, once I thought about it, something bothered me a little. Suppose is a real, finite-dimensional vector space. An inner product on is a symmetric, positive definite bilinear form . But any nondegenerate bilinear form yields an isomorphism to the dual. The map given by for any , is a map by ‘s linearit...
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