nForum - Search Results Feed (Tag: linear-algebra) 2021-04-12T08:25:39-04:00 https://nforum.ncatlab.org/ Lussumo Vanilla & Feed Publisher topological vector space https://nforum.ncatlab.org/discussion/6077/ 2014-07-09T09:02:24-04:00 2020-06-04T06:31:05-04:00 Urs https://nforum.ncatlab.org/account/4/ the first paragraphs at topological vector space seem odd to me. I’d think it should start out saying that a topological vector space is a vector space over a topological field kk, such that etc.. ...

the first paragraphs at topological vector space seem odd to me.

I’d think it should start out saying that a topological vector space is a vector space over a topological field $k$, such that etc.. Then the following remark presently in the entry, about the internalization using the discrete topology is moot.

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Vect(X) https://nforum.ncatlab.org/discussion/7789/ 2017-05-26T01:50:56-04:00 2020-06-04T06:30:17-04:00 Urs https://nforum.ncatlab.org/account/4/ Created a little entry Vect(X) (to go along with Vect) and used the occasion to give distributive monoidal category the Examples-section that it was missing and similarly touched the Examples-section ...

Created a little entry Vect(X) (to go along with Vect) and used the occasion to give distributive monoidal category the Examples-section that it was missing and similarly touched the Examples-section at rig category.

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super vector space https://nforum.ncatlab.org/discussion/7503/ 2016-11-16T07:32:37-05:00 2019-12-08T01:41:17-05:00 Urs https://nforum.ncatlab.org/account/4/ I gave the entry super vector space some expositional background and a more detailed (pedantic) definition.

I gave the entry super vector space some expositional background and a more detailed (pedantic) definition.

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Hilbert space https://nforum.ncatlab.org/discussion/5717/ 2014-03-01T00:26:45-05:00 2019-12-05T06:17:39-05:00 Urs https://nforum.ncatlab.org/account/4/ there had been no references at Hilbert space, I have added the following, focusing on the origin and application in quantum mechanics: John von Neumann, Mathematische Grundlagen der ...

there had been no references at Hilbert space, I have added the following, focusing on the origin and application in quantum mechanics:

• John von Neumann, Mathematische Grundlagen der Quantenmechanik. (German) Mathematical Foundations of Quantum Mechanics. Berlin, Germany: Springer Verlag, 1932.

• George Mackey, The Mathematical Foundations of Quamtum Mechanics A Lecture-note Volume, ser. The mathematical physics monograph series. Princeton university, 1963

• E. Prugovecki, Quantum mechanics in Hilbert Space. Academic Press, 1971.

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Fréchet space https://nforum.ncatlab.org/discussion/7262/ 2016-09-02T04:18:19-04:00 2019-04-12T20:51:10-04:00 Urs https://nforum.ncatlab.org/account/4/ At Fréchet space I have added to the Idea-section a paragraph motivating the definition via families of seminorms from the example of &Ropf; &infin;=lim&longleftarrow; n&Ropf; ...

At Fréchet space I have added to the Idea-section a paragraph motivating the definition via families of seminorms from the example of $\mathbb{R}^\infty = \underset{\longleftarrow}{\lim}_n \mathbb{R}^n$. And I touched the description of this example in the main text, now here.

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locally convex topological vector space https://nforum.ncatlab.org/discussion/8100/ 2017-10-23T07:06:30-04:00 2018-05-29T21:21:38-04:00 Urs https://nforum.ncatlab.org/account/4/ I have added to locally convex topological vector space the standard alternative characterization of continuity of linear functionals by a bound for one of the seminorms: here (proof and/or more ...

I have added to locally convex topological vector space the standard alternative characterization of continuity of linear functionals by a bound for one of the seminorms: here

(proof and/or more canonical reference should still be added).

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Kähler vector space and Hermitian space https://nforum.ncatlab.org/discussion/8203/ 2017-12-21T08:45:40-05:00 2017-12-21T08:45:40-05:00 Urs https://nforum.ncatlab.org/account/4/ I spelled out the elementary definitions, relations and examples at Kähler vector space and Hermitian space. This started out as a section that I added to Kähler manifold.

I spelled out the elementary definitions, relations and examples at Kähler vector space and Hermitian space.

This started out as a section that I added to Kähler manifold.

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direction of a vector https://nforum.ncatlab.org/discussion/8129/ 2017-11-06T05:48:42-05:00 2017-11-14T06:39:22-05:00 Urs https://nforum.ncatlab.org/account/4/ needed to be able to point to direction of a vector.

needed to be able to point to direction of a vector.

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wave vector https://nforum.ncatlab.org/discussion/8131/ 2017-11-06T05:57:53-05:00 2017-11-06T05:57:53-05:00 Urs https://nforum.ncatlab.org/account/4/ needed to be able to point to wave vector, so I created a bare minimum entry

needed to be able to point to wave vector, so I created a bare minimum entry

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inner product of vector bundles https://nforum.ncatlab.org/discussion/7790/ 2017-05-26T03:33:01-04:00 2017-05-26T08:46:38-04:00 Urs https://nforum.ncatlab.org/account/4/ created 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

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supersymmetry and division algebras https://nforum.ncatlab.org/discussion/7540/ 2016-12-14T13:22:58-05:00 2016-12-14T13:22:58-05:00 Urs https://nforum.ncatlab.org/account/4/ I have expanded the list of references at supersymmetry and division algebras.

I have expanded the list of references at supersymmetry and division algebras.

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quadratic refinement https://nforum.ncatlab.org/discussion/6009/ 2014-06-03T12:21:30-04:00 2016-09-15T10:25:28-04:00 Urs https://nforum.ncatlab.org/account/4/ I had splitt-off quadratic refinement from quadratic form and expanded slightly

I had splitt-off quadratic refinement from quadratic form and expanded slightly

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sesquilinear form https://nforum.ncatlab.org/discussion/7266/ 2016-09-06T10:08:57-04:00 2016-09-06T10:08:57-04:00 Urs https://nforum.ncatlab.org/account/4/ At sesquilinear form I have added the following general definition: let AA be a not-necessarily commutative star-algebra. Let VV be a left AA-module with AA-linear dual denoted V &ast;V^\ast. ...

At sesquilinear form I have added the following general definition:

let $A$ be a not-necessarily commutative star-algebra. Let $V$ be a left $A$-module with $A$-linear dual denoted $V^\ast$. Then a sesquilinear form on $V$ is simply an element in the tensor product

$V^\ast \otimes_A V^\ast \,,$

where we use the only possible way to regard the left $V$-module as a right $V$-module: by the star-involution.

I am wondering if there is anywhere some discussion as to how far one may push dg-algebra over $A$ this way, specifically for the cases where $A$ is a normed division algebra such that the quaternions or the octonions.

For instance which structure do we need on $A$ to make sense of the Grassmann algebra $\wedge^\bullet_A V^\ast$ of $V^\ast$ this way?

This is motivated by the following:

At spin representation I once put a remark that one may obtain the $N = 1$ super-translation Lie algebra simply by starting with the super-point $\mathbb{R}^{0\vert 2}$, regarded as an abelian super Lie algebra, and then forming the central extension by $\mathbb{R}^3$ which is classified by the cocycle $d \theta_i \wedge d \theta_j \in \wedge^2 (\mathbb{R}^2)^\ast$ $(1 \leq i \leq j \leq 2)$, with $\theta_i$ and $\theta_j$ the two canonical odd-graded coordinates on $\mathbb{R}^{0\vert 2}$.

Yesterday with John Huerta we were brainstorming about how to best formulate this such that the statement goes through verbatim for the other real normed division algebras to yield the super-translation Lie algebra alsoin dimensions 4,6 and 10.

With sesquilinear forms as above it is obvious: Let $\mathbb{K}$ any of the four real normed division algebras, consider the superpoint $\mathbb{K}^{0\vert 2}$ and then form the central extension of super Lie algebras classified by the sesquilinear forms $d \theta_i \otimes_{\mathbb{K}} d \theta_j$ ($1 \leq i \leq j \leq 2$).

These forms being sesquilinear expresses nothing but the spinor pairing of the susy algebra that Baez-Huerta (as reviewed here ) write as $(\psi,\phi)\mapsto \psi \phi^\dagger + \phi \psi^\dagger$.

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graded vector space https://nforum.ncatlab.org/discussion/6617/ 2015-05-28T12:21:57-04:00 2015-05-28T16:49:01-04:00 Urs https://nforum.ncatlab.org/account/4/ I see that (from long, long time ago) one section of the entry graded vector space defines “pre-graded” to mean &Zopf;\mathbb{Z}-graded and “graded” to be &Nopf;\mathbb{N}-graded. I ...

I see that (from long, long time ago) one section of the entry graded vector space defines “pre-graded” to mean $\mathbb{Z}$-graded and “graded” to be $\mathbb{N}$-graded.

I am not sure if that is a good terminology, mainly because it seems not to be common. I came here from the entry dg-Lie algebra, wondering what that entry might actually mean by a “pre-graded” Lie algebra. (I should have commented on this long ago, of course).

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differential operators as coKleisli morphisms for the Jet comonad https://nforum.ncatlab.org/discussion/6583/ 2015-05-12T12:28:45-04:00 2015-05-13T13:09:35-04:00 Urs https://nforum.ncatlab.org/account/4/ It seems to me that the category of bundles over some base space, with morphisms the differential operators on spaces of sections of these bundles, is equivalently the co-Kleisli category of the Jet ...

It seems to me that the category of bundles over some base space, with morphisms the differential operators on spaces of sections of these bundles, is equivalently the co-Kleisli category of the Jet bundle comonad. Is this known? (It seems to be a different statement than that discussed by Blute-Cockett-Seely, as far as I see.)

More in detail, consider differential cohesion with infinitesimal shape modality $\Im$. For a given base space $X$, write

$i \colon X \longrightarrow \Im X$

for the $X$-component of the unit of the $\Im$-monad. Then the operation of forming jet bundles is the comonad given by the base change adjoint triple $(i_! \dashv i^\ast \dashv i_\ast)$:

$Jet \coloneqq i^\ast i_\ast \;\colon\; \mathbf{H}_{/X} \to \mathbf{H}_{/X} \,.$

Now, it is a standard fact that given two bundles $E_1, E_2$ over $X$, then differential operators

$D \colon \Gamma(E_1) \to \Gamma(E_2)$

are equivalently bundle maps

$\tilde D \;\colon\; Jet(E_1) \longrightarrow E_2 \,,$

where the equivalence is given by

$D(\phi) = \tilde D \circ j^\infty(\phi)$

with $j^\infty \phi \in \Gamma(Jet(E_1))$ the jet bundle prolongation of $\phi$.

Combining this with the information that $Jet$ is a comonad, we have the impulse to say that $Jet(E_1) \to E_2$ is to be regarded as a morphism in its co-Kleisli category, and hence that under the above equivalence the composition of differential operators $D_2 \circ D_1$ corresponds to the composite

$Jet(E_1) \to Jet (Jet(E_1)) \stackrel{Jet(\tilde D_1)}{\longrightarrow} Jet(E_2) \stackrel{\tilde D_2}{\longrightarrow} E_3 \,,$

where the first map is the co-product of the $Jet$ co-monad.

And it seems to me that this is actually true.

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differential operator https://nforum.ncatlab.org/discussion/6584/ 2015-05-12T13:49:58-04:00 2015-05-12T14:26:18-04:00 Urs https://nforum.ncatlab.org/account/4/ added to differential operator the characterization via bundle maps out of a jet bundle, together with the note that this means that differential operators are equivalently morphisms in the ...

added to differential operator the characterization via bundle maps out of a jet bundle, together with the note that this means that differential operators are equivalently morphisms in the co-Kleisli category of the Jet bundle comonad.

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torsion module https://nforum.ncatlab.org/discussion/6192/ 2014-08-26T20:40:54-04:00 2014-08-26T20:40:54-04:00 Urs https://nforum.ncatlab.org/account/4/ just to clean up entries, I have given torsion module its own entry (the keyword used to have non-overlapping discussion at torsion subgroup and at torsion approximation)

just to clean up entries, I have given torsion module its own entry (the keyword used to have non-overlapping discussion at torsion subgroup and at torsion approximation)

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Fourier-Mukai transform https://nforum.ncatlab.org/discussion/6074/ 2014-07-04T15:30:10-04:00 2014-07-04T15:59:55-04:00 Urs https://nforum.ncatlab.org/account/4/ I gave Fourier-Mukai transform a bit of an Idea-section. It overlaps substantially with the Definition section now, but I thought one needs to say the simple basic idea clearly in words first. Also ...

I gave Fourier-Mukai transform a bit of an Idea-section. It overlaps substantially with the Definition section now, but I thought one needs to say the simple basic idea clearly in words first. Also added a few more pointers to literature.

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characteristic element of a bilinear form https://nforum.ncatlab.org/discussion/6004/ 2014-06-02T17:44:08-04:00 2014-06-02T17:44:08-04:00 Urs https://nforum.ncatlab.org/account/4/ created characteristic element of a bilinear form ]]> twisted generalized cohomology in linear homotopy-type theory https://nforum.ncatlab.org/discussion/5651/ 2014-02-04T14:42:22-05:00 2014-02-04T14:42:22-05:00 Urs https://nforum.ncatlab.org/account/4/ I’d be trying to write out a more detailed exposition of how fiber integration in twisted generalized cohomology/twisted Umkehr maps are axiomaized in linear homotopy-type theory. To start with I ...

I’d be trying to write out a more detailed exposition of how fiber integration in twisted generalized cohomology/twisted Umkehr maps are axiomaized in linear homotopy-type theory.

To start with I produced a dictionary table, for inclusion in relevant entries:

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superoperator https://nforum.ncatlab.org/discussion/5617/ 2014-01-21T02:46:55-05:00 2014-01-21T02:46:55-05:00 Urs https://nforum.ncatlab.org/account/4/ brief entry superoperator, just for completeness

brief entry superoperator, just for completeness

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preconditioner https://nforum.ncatlab.org/discussion/5537/ 2013-12-05T18:38:25-05:00 2013-12-05T18:38:25-05:00 zskoda https://nforum.ncatlab.org/account/10/ preconditioner

preconditioner

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real structure https://nforum.ncatlab.org/discussion/5210/ 2013-08-29T20:37:00-04:00 2013-08-29T20:37:00-04:00 Urs https://nforum.ncatlab.org/account/4/ I had had need to link to and hence create some trivial entries, such as real structure and antilinear map. Didn’t find time yet to fill anything non-stubby into quaternionic structure.

I had had need to link to and hence create some trivial entries, such as real structure and antilinear map. Didn’t find time yet to fill anything non-stubby into quaternionic structure.

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(co)isotropic subspaces - table https://nforum.ncatlab.org/discussion/4796/ 2013-03-18T20:17:47-04:00 2013-03-18T20:17:47-04:00 Urs https://nforum.ncatlab.org/account/4/ created a simple table (co)isotropic subspaces - table for inclusion in other entries, just so as to usefully cross-link the relevant entries

created a simple table (co)isotropic subspaces - table for inclusion in other entries, just so as to usefully cross-link the relevant entries

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hyperdeterminant https://nforum.ncatlab.org/discussion/4663/ 2013-01-10T16:06:14-05:00 2013-01-10T16:06:14-05:00 zskoda https://nforum.ncatlab.org/account/10/ New stub hyperdeterminant (I was convinced we had it before, but…no).

New stub hyperdeterminant (I was convinced we had it before, but…no).

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