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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeMay 26th 2017
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   Author: Urs
   Format: MarkdownItexcreated _[[inner product of vector bundles]]_ with the construction over paracompact Hausdorff spaces

   created inner product of vector bundles with the construction over paracompact Hausdorff spaces

2. • CommentRowNumber2.
   • CommentAuthorOscar_Cunningham
   • CommentTimeMay 26th 2017
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   Author: Oscar_Cunningham
   Format: MarkdownItexI think we should change $E\oplus_X E$ to $E\times_X E$ and $E_x\oplus E_x$ to $E_x\times E_x$. In my mind $V\oplus V$ is a vector space whereas $V\times V$ is a just set. Writing $E_x\oplus E_x\to\mathbb{R}$ makes me think that the map is linear, whereas $E_x\times E_x\to\mathbb{R}$ would just be a function. This is how it's done on the page "[inner product space](https://ncatlab.org/nlab/show/inner+product+space)", it uses $\times$ rather than $\oplus$.

   I think we should change $E\oplus_X E$ to $E\times_X E$ and $E_x\oplus E_x$ to $E_x\times E_x$. In my mind $V\oplus V$ is a vector space whereas $V\times V$ is a just set. Writing $E_x\oplus E_x\to\mathbb{R}$ makes me think that the map is linear, whereas $E_x\times E_x\to\mathbb{R}$ would just be a function. This is how it’s done on the page “inner product space”, it uses $\times$ rather than $\oplus$.

3. • CommentRowNumber3.
   • CommentAuthorDmitri Pavlov
   • CommentTimeMay 26th 2017
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   Author: Dmitri Pavlov
   Format: MarkdownItexWouldn't it be better to simply use ⊗?

   Wouldn’t it be better to simply use ⊗?

4. • CommentRowNumber4.
   • CommentAuthorTodd_Trimble
   • CommentTimeMay 26th 2017
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   Author: Todd_Trimble
   Format: MarkdownItexWell, you definitely don't want to say "vector bundle map $E \oplus_X E \to X \times \mathbb{R}$", 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'.

   Well, you definitely don’t want to say “vector bundle map $E \oplus_X E \to X \times \mathbb{R}$”, 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’.

5. • CommentRowNumber5.
   • CommentAuthorUrs
   • CommentTimeMay 26th 2017
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   Author: Urs
   Format: MarkdownItexWoops, sorry. My mind was still on the entry on direct sum of vector bundles, it seems. Fixed now.

   Woops, sorry. My mind was still on the entry on direct sum of vector bundles, it seems. Fixed now.

6. • CommentRowNumber6.
   • CommentAuthorDavidRoberts
   • CommentTimeMay 26th 2017
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   Author: DavidRoberts
   Format: MarkdownItexAlso, it's not an inner product _of_ vector bundles, rather _on_ vector bundles. I was expecting some sort of categorified thing $Vect(X)\times Vect(X) \to $???

   Also, it’s not an inner product of vector bundles, rather on vector bundles. I was expecting some sort of categorified thing $Vect(X)\times Vect(X) \to$???

7. • CommentRowNumber7.
   • CommentAuthorUrs
   • CommentTimeMay 26th 2017
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   Author: Urs
   Format: MarkdownItexGood point, I have changed "of" to "on".

   Good point, I have changed “of” to “on”.