Articles of duality theorems

Examples of double dual spaces

I am looking for examples on double dual spaces. As I know $\ell_p $ is the double dual of $\ell_p$ for $1<p,q<\infty$, $\mathcal L_p $ is the double dual for $\mathcal L_p$ for $1<p,q<\infty$ and $\ell_\infty$ is the double dual of $c_0$, where $c_0$ is the space of sequences of numbers that converge to 0. […]

Image of dual map is annihilator of kernel

Suppose $T:V\to W$ and that $V$ is finite-dimensional. I want to prove that $$\text{Im }T’=(\ker T)^0$$ where $T’$ is the dual/transpose map and $(\ker T)^0$ is the annihilator of the kernel. I know that $\phi \in V’$ is an annihilator of $\ker T$ if and only if $$\phi(v)=0 \space \forall v\in \ker T$$ if and […]

If this problem is not unbounded, what's wrong in this dual derivation?

In a paper with 100 citation, Robust Support Vector Machine Training via Convex outlier Ablation, a convex relaxation is used. In this paper, a form of robust svm proposed: \begin{align} \min_{0\leq \eta \leq e} \min_{w,\xi} &\frac{\beta}{2}\Vert w\Vert^2 + e^T \xi+ e^T (e-\eta)\cr \text{s.t} &\xi \geq 0, \xi \geq diag(\eta) (e-YX^T w)\cr =\min_{0\leq \eta \leq e}\max_{0\leq […]

Pontryagin dual of the unit circle

Define the unit circle as $\frac{\mathbb{R}}{2\pi\mathbb{Z}}.$ I know the Pontryagin dual (looking at properties of the Fourier transform on locally compact Abelian groups) is $\mathbb{Z}$ but why? Any notes or suggestions will be appreciated.

Consequences of cycle space cut space duality

The cycle space cut space duality theorem for planar graphs states that: The cycle space of a planar graph is the cut space of its dual graph, and vice versa. I wish to know any consequences and applications of this result with possible proofs or references to them.

Representation of a linear functional Lipschitz in total variation

Let $\Omega$ be a Borel space and let $\mathcal P(\Omega)$ be the space of all Borel probability measures on $\Omega$ endowed with the topology of weak convergence. Define the total variation metric on the latter space $$ d(\mu,\nu) :=\sup\left\{\int_\Omega f\;\mathrm d\mu – \int_\Omega f\;\mathrm d\nu:f\in \mathcal B_1(\Omega)\right\} $$ where $\mathcal B_1(\Omega)$ is the space of […]

How to test if a feasible solution is optimal – Complementary Slackness Theorem – Linear Programming

I have this linear program $$\begin{cases} \text{max }z=&5x_1+7x_2-3x_3\\ &2x_1+4x_2-2x_3&\le8\\ &-x_1+x_2+2x_3&\le10\\ &x_1+2x_2-x_3&\le6\\ &x_1,\,x_2,\,x_3\ge0 \end{cases}$$ and a feasible solution of his dual $(D)$ is $y = {7/2,2,0}$. I need to find an optimal basis of $(P)$ and an optimal basis of $(D)$ using the complementary slackness theorem. I thought about assuming that $y$ is an optimal solution […]

Concerning $f(x_1, \dots , x_n)$

I am not getting even an intuition as how to solve this problem. Please help me with a solution. Let $n$ be a positive integer and $F$ a field. Let $W$ be the set of all vectors $(x_1, \dots , x_n)$ in $F^n$ such that $x_1+\dots +x_n =0$. $1)$ Prove that $W^0$ (annihilator of $W$) […]

If a normed space $X$ is reflexive, show that $X'$ is reflexive.

If a normed space $X$ is reflexive, show that $X’$ is reflexive. Suppose $X$ is reflexive. Then by definition the Canonical mapping $J : X \to X”$ defined by $x \mapsto g_x$ where $g_x(f) = f(x)$ is an isomorphism. We want to show that the mapping $J’ : X’ \to X”’$ defined by $f \mapsto […]

Which concepts in Differential Geometry can NOT be represented using Geometric Algebra?

1. It is not clear to me that linear duals, and not just Hodge duals, can be represented in geometric algebra at all. See, for example, here. Can linear duals (i.e. linear functionals) be represented using the geometric algebra formalism? 2. It also seems like most tensors cannot be represented, see for example here. This […]