Questions tagged [dual-pairs]
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11 questions
2
votes
2
answers
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Dual space of the intersection of locally convex vector spaces
Let $S \neq \emptyset$ and let $\big((E_s,\mathcal{T}_s)\big)_{s \in S}$ be a family of locally convex vector subspaces of the same vector space. Denote by $E_s^*$ the dual space of $(E_s,\mathcal{T}...
- 123
4
votes
1
answer
182
views
Subgradient in a predual under weak* continuity
Let $X$ be a Banach space. Suppose $f:X^*\to\mathbb R\cup\{\infty\}$ is convex, has weak*-compact effective domain, and is weak*-continuous on its effective domain. In particular, $f$ is weak*-lower ...
- 537
5
votes
1
answer
506
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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 ...
3
votes
0
answers
650
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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 ...
- 325
1
vote
1
answer
1k
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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 ...
- 119
28
votes
4
answers
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 ...
- 6,919
57
votes
6
answers
6k
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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 ...
- 3,511
10
votes
2
answers
1k
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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 ...
- 33.3k
5
votes
3
answers
5k
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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 ...
- 151
6
votes
2
answers
1k
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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 ...
- 957
5
votes
2
answers
1k
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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 ...
- 605