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1. • CommentRowNumber1.
   • CommentAuthorTobyBartels
   • CommentTimeAug 15th 2012
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   Author: TobyBartels
   Format: MarkdownItexNew article: [[direct sum of Banach spaces]]. These come in even more variety than I originally thought!

   New article: direct sum of Banach spaces. These come in even more variety than I originally thought!

2. • CommentRowNumber2.
   • CommentAuthorTodd_Trimble
   • CommentTimeAug 15th 2012
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   Author: Todd_Trimble
   Format: MarkdownItexDisambiguation? It seems the direct sums being considered in that article are not (usually) direct sums in the sense of [[direct sum]]. Could be worth pointing out that the $l^2$ direct sum of Hilbert spaces coincides with the coproduct in the category of Hilbert spaces (at least, I think that's true).

   Disambiguation? It seems the direct sums being considered in that article are not (usually) direct sums in the sense of direct sum.

   Could be worth pointing out that the $l^2$ direct sum of Hilbert spaces coincides with the coproduct in the category of Hilbert spaces (at least, I think that’s true).

3. • CommentRowNumber3.
   • CommentAuthorTobyBartels
   • CommentTimeAug 16th 2012
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   Author: TobyBartels
   Format: MarkdownItexI'm not sure disambiguation is needed; both articles refer to each other.

   I’m not sure disambiguation is needed; both articles refer to each other.

4. • CommentRowNumber4.
   • CommentAuthorTobyBartels
   • CommentTimeAug 16th 2012
   • PermaLink
   Author: TobyBartels
   Format: MarkdownItexYou're right about Hilbert spaces; I put that in, and more.

   You’re right about Hilbert spaces; I put that in, and more.

5. • CommentRowNumber5.
   • CommentAuthorTodd_Trimble
   • CommentTimeAug 16th 2012
   • (edited Aug 16th 2012)
   • PermaLink
   Author: Todd_Trimble
   Format: MarkdownItexRe #3: well, a warning then, along the lines that a "direct sum of Banach spaces is not the same as a direct sum in the category of Banach spaces". Edit: okay, I saw that you had remarks pertaining to this under the Properties section. I made some slight changes in the Idea section, to make less likely a misunderstanding that direct sums in the sense being considered here are the same as [[direct sum]]s in the category of Banach spaces and short linear maps.

   Re #3: well, a warning then, along the lines that a “direct sum of Banach spaces is not the same as a direct sum in the category of Banach spaces”.

   Edit: okay, I saw that you had remarks pertaining to this under the Properties section. I made some slight changes in the Idea section, to make less likely a misunderstanding that direct sums in the sense being considered here are the same as direct sums in the category of Banach spaces and short linear maps.

6. • CommentRowNumber6.
   • CommentAuthorTobyBartels
   • CommentTimeAug 16th 2012
   • PermaLink
   Author: TobyBartels
   Format: MarkdownItexI see; yes, what I had there was confusing.

   I see; yes, what I had there was confusing.

7. • CommentRowNumber7.
   • CommentAuthorTobyBartels
   • CommentTimeAug 16th 2012
   • PermaLink
   Author: TobyBartels
   Format: MarkdownItexAlso [[tensor product of Banach spaces]], which needs details (but links to Wikipedia, which has some).

   Also tensor product of Banach spaces, which needs details (but links to Wikipedia, which has some).

8. • CommentRowNumber8.
   • CommentAuthorTobyBartels
   • CommentTimeAug 18th 2012
   • PermaLink
   Author: TobyBartels
   Format: MarkdownItexThe tensor product article is done (I mean as much as I'm likely to do).

   The tensor product article is done (I mean as much as I’m likely to do).

9. • CommentRowNumber9.
   • CommentAuthorTodd_Trimble
   • CommentTimeAug 18th 2012
   • PermaLink
   Author: Todd_Trimble
   Format: MarkdownItexNitpick perhaps, but in the first displayed line of Definition 1, there's no tensor product symbol between the $v$'s and $w$'s. Is there a reason for that? (To me, it looks slightly confusing.)

   Nitpick perhaps, but in the first displayed line of Definition 1, there’s no tensor product symbol between the $v$’s and $w$’s. Is there a reason for that? (To me, it looks slightly confusing.)

10. • CommentRowNumber10.
    • CommentAuthorTobyBartels
    • CommentTimeAug 18th 2012
    • PermaLink
    Author: TobyBartels
    Format: MarkdownItexWhen I put it in there, it seemed cluttered and unnecessary. (It *is* unnecessary, since we're only forming formal products, and one can use whatever symbol one likes. In contrast, in $\lambda \otimes \mu$ in Definition 2, this is not a formal product but a specific operation.) It would look less cluttered if things were more symmetric; if the $\alpha$s weren't there (being absorbed into the $v$s or $w$s). But I like having $\alpha$ conceptually, since otherwise you get the wrong thing for the $0$-ary tensor product. Anyway, put the symbol in if you like.

    When I put it in there, it seemed cluttered and unnecessary. (It is unnecessary, since we’re only forming formal products, and one can use whatever symbol one likes. In contrast, in $\lambda \otimes \mu$ in Definition 2, this is not a formal product but a specific operation.) It would look less cluttered if things were more symmetric; if the $\alpha$s weren’t there (being absorbed into the $v$s or $w$s). But I like having $\alpha$ conceptually, since otherwise you get the wrong thing for the $0$-ary tensor product.

    Anyway, put the symbol in if you like.

11. • CommentRowNumber11.
    • CommentAuthorTodd_Trimble
    • CommentTimeAug 18th 2012
    • PermaLink
    Author: Todd_Trimble
    Format: MarkdownItexYes, I see your reasoning. I left it out, but made a note just prior, just to set at ease any minds that might need easing.

    Yes, I see your reasoning. I left it out, but made a note just prior, just to set at ease any minds that might need easing.

12. • CommentRowNumber12.
    • CommentAuthorTobyBartels
    • CommentTimeAug 18th 2012
    • PermaLink
    Author: TobyBartels
    Format: MarkdownItexOK. FWIW, I suppressed the symbol in Definition 3 as well; but when I came back a day later to write Definition 4, I kept it!

    OK. FWIW, I suppressed the symbol in Definition 3 as well; but when I came back a day later to write Definition 4, I kept it!