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Questions tagged [dual-pairs]

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4
votes
0answers
75 views

Generalized Gelfand Triples

Normally, when we talk about Gelfand triple we have three Hilbert spaces $$\newcommand{\X}{\mathcal{X}} \X_+ \subset \X_0 \subset \X_- $$ such that the subsets are dense, the embedding mappings are ...
2
votes
0answers
86 views

description of dual space of space of Radon measure equipped with topology of weak convergence

Let $\mathcal{M}(\mathbb R)$ be the space of Radon measures, equipped with topology $\tau$ generated by the following "weak convergence": $$ \mu_n \rightarrow \mu \quad \text{iff} \quad \int f d\mu_n ...
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vote
1answer
811 views

Question on convex optimization and dual norms [closed]

I have the following questions about dual norms : How do you prove that the dual of the dual norm is in fact the original norm? This is what I have so far: If I have $||y||_* $ as the norm dual of ...
26
votes
4answers
2k views

Matrices: characterizing pairs $(AB, BA)$

Let $A$ be an $m\times n$-matrix and $B$ an $n \times m$-matrix over the same field. Consider the matrices $C=AB$ and $D=BA$. It is probably well known (and not difficult to show) that the only ...
45
votes
6answers
3k views

Is the non-triviality of the algebraic dual of an infinite-dimensional vector space equivalent to the axiom of choice?

If $V$ is given to be a vector space that is not finite-dimensional, it doesn't seem to be possible to exhibit an explicit non-zero linear functional on $V$ without further information about $V$. The ...
9
votes
2answers
761 views

How to decompose a composition of representations?

Background I would like to know if there is some slick machinery to solve the following representation-theoretic problem. Let $\left(V,\langle-,-\rangle\right)$ be a finite-dimensional real inner ...
4
votes
3answers
3k views

Spherical Harmonics - a bunch of questions about them

Hi there, Please tell me if I should divide these into individual questions next time. Short intro: Spherical Harmonics are a nice collection of functions. They are orthogonal and allow you to take ...
6
votes
2answers
885 views

Which tensor fields on a symplectic manifold are invariant under all Hamiltonian vector fields?

Consider a connected symplectic manifold $(M, \omega)$ of dimension $m=2n$. A few preliminary reminders (mostly to fix the notation): A vector field $X$ is symplectic if its flow preserves the ...
5
votes
2answers
748 views

Howe duality for exceptional algebras

There is a nice tool in representation theory, the Howe duality, which as I know works for certain pairs of classical Lie algebras (the reference to the complete list of Howe dual pairs is appreciated ...