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Banach spaces, function spaces, real functions, integral transforms, theory of distributions, measure theory.

12 votes
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Horn's inequalities for n matrices

1) Section 7 of [Knutson-Tao-Woodward], http://arxiv.org/pdf/math/0107011.pdf 2) Yes. To obtain the (minimal set of) inequalities, you can just glom the puzzles together.
Allen Knutson's user avatar
17 votes
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Why do inner products require conjugation?

Bi- (or sesqui-) linear forms are nicer if they're nondegenerate. But they can always be restricted to subspaces. So, they're even nicer if they're nondegenerate on all subspaces. For symmetric forms …
Allen Knutson's user avatar