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created inner product of vector bundles with the construction over paracompact Hausdorff spaces
I think we should change E⊕XE to E×XE and Ex⊕Ex to Ex×Ex. In my mind V⊕V is a vector space whereas V×V is a just set. Writing Ex⊕Ex→ℝ makes me think that the map is linear, whereas Ex×Ex→ℝ would just be a function. This is how it’s done on the page “inner product space”, it uses × rather than ⊕.
Wouldn’t it be better to simply use ⊗?
Well, you definitely don’t want to say “vector bundle map E⊕XE→X×ℝ”, because that’s just wrong. I’d think it best to follow Dmitri’s advice, and then break it down into bilinear maps if that seems too high-falutin’.
Woops, sorry. My mind was still on the entry on direct sum of vector bundles, it seems. Fixed now.
Also, it’s not an inner product of vector bundles, rather on vector bundles. I was expecting some sort of categorified thing Vect(X)×Vect(X)→???
Good point, I have changed “of” to “on”.
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