# Tagged Questions

For question about the concept of dual, either in the sense of vector spaces, topological group or dual problems.

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### Prob. 8, Sec. 2.10 in Erwine Kreyszig's INTRODUCTORY FUNCTIONAL ANALYSIS WITH APPLICATIONS: The dual space of $c_0$ is $\ell^1$?

Let $c_0$ be the subspace of $\ell^\infty$ consisting of all sequences of (real or complex ) numbers converging to $0$. How to prove that the dual space of $c_0$ is (isomorphic to) $\ell^1$? My ...
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### Cohomology with compact support for sheaves in separated schemes of finite type over a Noetherian scheme: three different definitions

usually there are three notions of cohomology with compact (proper) support. The first one usually done in the Ã©tale site. However the second one is used in Verdier duality. The third one is done in ...
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### What decides the structure of the dual variables taken in designing min-max type combinatorial optimization algorithms?

There are a bunch of combinatorial optimization problems like min cost flows and min weight perfect matchings that invoke duality and complimentary slackness to improve the primal feasible solution. ...
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### Duality of Linear Program

Formulate the dual of the following linear problem: Min $\sum$ $\sum$ $c_{ij}$ $x_{ij}$ s.to $\sum$ $x_{ij}$ - $\sum$ $x_{ji}$ = 0, $\hspace{5mm}$ $x_{ij}$ >= $l_{ij}$ and $x_{ij}$ $\leq$ $u_{ij}$ ...
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### Dual Decomposition with multiple coupling constraints

This is probably a a simple question, but have been stuck on this for a while and unable to figure out my issue from the standard Boyd/Vandhenbergen decomposition references. I am interested in dual ...
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### Dual of $L^p$ when $p = 0$?

I've spent some time searching for this online - both on this site and elsewhere - and even after consulting a considerable amount of literature, I can't seem to nail down an answer. Perhaps someone ...
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### Proving that two problems are strongly dual when solutions are restricted to a space

Consider the following problems with solutions $\mathbf{w}\in\mathbb{R}_{++}^n$ \begin{align} (P) \hspace{.3in} \min_{\mathbf{w}} \hspace{.3cm} & \mathbf{p}^H\cdot\mathbf{w} \\ \text{s.t. } & ...
I am familiar with Rudin's book's proof of the fact that, in $\sigma$-finite measure spaces and for $p\in[1,+\infty)$, the dual space of $L^p$ is $L^q$ where $p,q$ are conjugate, i.e. ...