The dual-pairs tag has no usage guidance.

**1**

vote

**1**answer

296 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 ...

**22**

votes

**4**answers

1k 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 ...

**34**

votes

**6**answers

2k 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 ...

**8**

votes

**2**answers

602 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

**3**answers

2k 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 ...

**5**

votes

**2**answers

748 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

**2**answers

569 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 ...