Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

• Username
• Password
• Remember me

• Sign in using OpenID
• Remember me

• Forgot your password?
• Apply for membership

2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry book bundles calculus categorical categories category category-theory chern-weil-theory cohesion cohesive-homotopy-type-theory cohomology colimits combinatorics complex complex-geometry computable-mathematics computer-science constructive cosmology deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality elliptic-cohomology enriched fibration foundation foundations functional-analysis functor gauge-theory gebra geometric-quantization geometry graph graphs gravity grothendieck group group-theory harmonic-analysis higher higher-algebra higher-category-theory higher-differential-geometry higher-geometry higher-lie-theory higher-topos-theory homological homological-algebra homotopy homotopy-theory homotopy-type-theory index-theory integration integration-theory internal-categories k-theory lie-theory limits linear linear-algebra locale localization logic mathematics measure measure-theory modal modal-logic model model-category-theory monad monads monoidal monoidal-category-theory morphism motives motivic-cohomology nlab noncommutative noncommutative-geometry number-theory of operads operator operator-algebra order-theory pages pasting philosophy physics pro-object probability probability-theory quantization quantum quantum-field quantum-field-theory quantum-mechanics quantum-physics quantum-theory question representation representation-theory riemannian-geometry scheme schemes set set-theory sheaf simplicial space spin-geometry stable-homotopy-theory stack string string-theory superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory tqft type type-theory universal variational-calculus

1. • CommentRowNumber1.
   • CommentAuthorTobyBartels
   • CommentTimeAug 15th 2012
   • PermaLink
   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
   • PermaLink
   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
   • PermaLink
   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!