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

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Duality gap analysis

I solved a non-linear non-convex optimization problem via dual decomposition optimization using sub-gradient method. (my main goal is to solve the problem in a distributed way). I solve the same ...
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8 views

Application of projective duality

What are the applications of the projective duality principle in another scientific areas such as Physics and Chemistry?
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2answers
16 views

Example of a weakly separable normed space that is not norm separable

I am looking for an example of a normed space which is separable with respect to the topology induced by all continuous linear functionals, but not separable with respect to the norm topology.
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18 views

Self dual curves

Can you please tell me some of the applications of self dual curves or duality of any curves as a whole,because I can't find any of these, even though I've been looking very hard.
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13 views

Discrete mathematics: propositional calculus [closed]

Please state and explain the duality law and De Morgan's theorem for propositional calculus
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what is the dual problem of finding chebyshev's center?

let $a_j$ $(j=1,...,m)$ be a set of points in $R^2$. The problem of finding Chebyshev's center is: min r s.t. $norm(x-a_j)<=r$ $(j=1,...,m)$ Where r is the maximal radius and x is the center ...
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19 views

Max-flow-min-cut using LP duality

https://www.cs.oberlin.edu/~asharp/cs365/papers/Approximation-ch12.pdf is a chapter from Vazirani that discusses max cut-min flow using LP duality. The binary min-cut problem is: \begin{align} ...
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1answer
23 views

Showing that compact metrizable abelian group has enough characters

I wanted to know if anybody has some clue as why a compact metrizable abelian group has enough characters to sepatare points. (a character on a topological group is a continuous homomorphism from the ...
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13 views

Inner product space of continuous functions, showing inexistence

Let $E$ be the inner product theorem of the Continuous functions over the interval $[a,b]$, with the inner product $(x,y)=\int_a^b x(t)y(t)dt$. Fix $c\in [a,b]$, and let $f_c\in E^*$ such that ...
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31 views

Characterization of Bochner dual

I want to prove following theorem Let X be separable and reflexive Banach space, $1<p<\infty$ than $$ L^p((0,1),X)^* = L^q((0,1),X^*) $$ where $\frac1{p}+\frac1{q} = 1$, with ...
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38 views

What is the right (a good) definition for a dual monoid?

Suppose we have the free abelian monoid $S = \{a^m : m \in \mathbb{N}_0\}$ on the set of one element $X = \{a\}$. The binary operation on the monoid is denoted by $\cdot$. If $(T,\ast)$ is another ...
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1answer
33 views

Two duality theorems

Suppose $X$ is a Hilbert space with norm $||.||$ and $K$ is a weak compact and convex subset of $X$. The supporting functional: $$h(x^*)=\sup_{x\in K} \langle x^*, x \rangle$$ The indicator ...
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21 views

Conic Optimization

In a recent paper, Conice Optimization via Operator Splitting and Homogeneous Self-Dual Embedding, a primal of the form \begin{alignat}{3} &\text{minimize} &&c^T x\cr ...
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1answer
52 views

How to prove that $(C[a,b], \|\cdot\|_\infty)$ is not a reflexive Banach Space [duplicate]

The tag line basically says it all...this is a question in Luenberger's Optimization book (5.14.4 on p.138). Clearly I don't expect someone to deliver a full proof if it's tedious, but a sketch or ...
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32 views

Proof of Alexander Duality on Bredon - help with a passage

At page 353 of Bredon's Topology and Geometry, there is stated the Alexander Duality as Corollary $8.7$. I don't understand where does the upper row come from and why is it exact. I thought it was ...
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1answer
41 views

Gradient of a Lagrange dual function

Consider: $$\min_{x \in \mathbb{R}^n} f(x)$$ $$\ \ \ \ \ \ \ \text{s.t. }\ h(x) \leq 0$$ Lagrangian:$\ \ \ L(x,\lambda) = f(x) + \lambda h(x)$ Suppose $x^* = \arg\min_{x} L(x,\lambda)$ ...
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2answers
65 views

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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30 views

Filter,dual ideal,definition of $\liminf_I \lambda$

We have this http://shelah.logic.at/files/506.pdf definition of $\lim \inf_I \bar{\lambda}$ in 1.1(3): for $I$ a filter on $\kappa$ let $I^+=2^\kappa \setminus I$.$$\lim \inf_I ...
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1answer
28 views

Is it necessary to use the Hahn-Banach theorem to show that $(X/M)^*\simeq M^\perp$?

Let $X$ be a Banach space with dual space $X^*$, and let $M$ be a closed subspace of $X$. Then $M^\perp=\{x^*\in X^*: x*(m)=0 \text{ for all } m\in M\}$ is a closed subspace in $X^*$. I read the ...
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1answer
45 views

Prove that the dual -space of the dual-space of V is isomorphic to V without using bases

Given a vector space $V$ the dual space $V^*$ is the space of all linear operators from $V$ to $\mathbb{C}$. $V^*$ is itself a vector space and I know how to prove $V \cong (V^*)^*$ by using a ...
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2answers
36 views

Proof that $V^*$ is isomorphic to $V$.

In my notes for a linear algebra course there is proof that $V^*$ is isomorphic to $V$. However I am unclear on a few of the steps. We begin by choosing a basis $B = \{v_1,...,v_n\}$ for $V$. We now ...
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22 views

A question involving duality maps

Show that if X is an infinit dimensional and smooth Banach space, then there are no compact duality maps on X. Can someone, please, give me a hint on how to deduce this from the following fact: Let ...
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2answers
63 views

How to compute primal variable based on dual variables and their multipliers

I edited this question based on information I got from comments. Assume we have an optimization problem (primal problem). we solve it's dual using some kind of primal-dual interior point solver. So, ...
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1answer
30 views

Is the normalized duality mapping symmetric?

In the area of functional analysis, nonlinear operator theory, the normalized duality mapping on a Banach space $X$ is a set valued map from $J:X\rightarrow 2^{X^*}$ given by \begin{align} ...
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Equivalence of categories ($c^*$ algebras <-> topological spaces)

I try to use a littlebit category theory to have a better overview of the results in the theory of $c^*$-algebras, but I really have to read an introduction to category theory because I know almost ...
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61 views

Does this hold for $p=\infty $, i.e., is it true that $(l^{\infty})'= l^1? $ [closed]

Let $E=l^p$ where $1 \le p < \infty $ we know $E'=l^q$ Where $q$ is the dual exponent of $p$, i.e. $q$ is such that $\frac{1}{p}+\frac{1}{q}=1$ Does this hold for $p=\infty $, i.e., is it true ...
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2answers
48 views

Some lengthy question on natural transformations, category theory, and dual objects

So I am fairly new to algebra, and my instructor often uses terms from category theory (such as Hom-set, natural transformations, etc) that I am not familiar with. As an attempt to have a better ...
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69 views

Three theorems for the price of one? (like duality)

Why the notion of "duality" (when we get two theorems for the price of one) are ubiquitous in mathematics (order and lattice theory, category theory, group theory), but "triality" (three theorems for ...
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1answer
36 views

Is a finite cyclic group a Poincare duality group?

I am trying to understand whether the finite cyclic group of order $n$, $C_n$ is a Poincare duality group, i.e. whether it's classifying space $K(C_n,\,1)$ is a Poincare complex. I know that the ...
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1answer
48 views

Prove there exists a unique map T

Let $V_i$ be a collection of vector spaces over field $F$ where $i=1,2,...,N$. Given the Cartesian Product $V=V_1\times V_2\times...\times V_N$ equipped with natural projections $p_i:V\to V_i$. ...
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1answer
53 views

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? ...
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1answer
28 views

The dual of transporting problem

So basically I'm trying to figure out what does a certain variable in dual of transporting problem mean. Transporting problem in matrix form: (We are searching for a min cost of transferring goods ...
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1answer
35 views

Does every closed subspace of a dual space correspond to a closed subspace of its predual?

Suppose $X$ is a Banach space with dual space $X^*$. If $Y$ is a closed subspace of $X$, then $Y^\perp=\{x^*\in X^*: x^*(y)=0 \text{ for all } y\in Y\}$ is a closed subspace in $X^*$. I am wondering ...
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1answer
36 views

Duality (conjugate) function for $f \in L^\infty$

For $f\in L^\infty(\mathbb{R})$, can I find the $f^*\in (L^\infty(\mathbb{R}))^*$ such that $$\|f^*\|_* = 1 \text{ and } \langle f^*, f \rangle = \|f\|_\infty.$$ I know that ...
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37 views

Sufficient to check intersection of sub base elements with a dense set in compact open topology

I am reading a book about duality and there was this following claim: If I have a compact group G* (dual group of G, and G is discrete) with the compact open topology, then for any A, a subset of G* ...
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Question about duality in nonlinear optimization

Let $f(x)$ and $h(x)$ be functions from $\mathbb{R}^n$ to $\mathbb{R}$ and consider the minimization problem $$ {\rm minimize} ~~~ f(x)$$ $$~~~~~~~~~{\rm subject ~to}~~h(x)=0.$$ Suppose the minimum is ...
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1answer
31 views

Why is the determinant of a basis of a dual space non zero?

Let $P[t]{_2}$ = $V$ a vector space. A basis $B$ = $(1,t,t^2)$ of V and $B$* = ($e_1$,$e_2$,$e_3$) the dual basis of $B$. $f_a$: $V$ $->$ $R$ , $p(t)$ $->$ $p(a)$ (evaluation). Show that ...
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17 views

base of open neighborhood for dual group in k-topology

I wanted to ask the following: Suppose I have an abelian topological $G$, and $G^*$ is its dual group (all the continuous homomorphisms from $G$ to the circle group $T$). How can I show that the ...
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2answers
30 views

Showing that a dual basis is a generating set

Suppose we have a dual basis $F$* = ($f_1$*,......$f_n$*) of $V$*. and suppose the standard basis $F$ = ($f_1$,......$f_n$) of $V$. I want to show that the $F$* is a basis so i have to show that it is ...
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60 views

Getting explicit expression of function from the dual function

Considering the following problem : $$ minimize_{x_1,x_2} \ -2x_1+x_2 \\ subject \ to \ x_1+x_2=\frac52 \\ (x_1,x_2) \in X ,\\$$ where $X=\{(0,0),(0,2),(2,0),(2,2),(\frac54,\frac54)\}$ The dual ...
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23 views

Dual group endowed with the compact-open topology

I wanted to ask a question. Let $G^*$ be the dual group of an abelian topological group $G$ ($G^*$ is defined to be the group of all continuous homomorphisms from $G$ to the circle group $T$). I ...
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52 views

Example of Hopf categories and conatural transformation

Let $\mathcal{C}$ be a category. It is well known how to internalize the notion of category. Let $(C_0,C_1)$ be an internal category, with source $s$, target $t$, composition $c$ and unit $e$. One can ...
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1answer
130 views

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 ...
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1answer
100 views

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 étale site. However the second one is used in Verdier duality. The third one is done in ...
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1answer
29 views

Gradient of this function

I am to solve an optimization problem as described below: $$ \min f(x) = \frac{1}{2}\left\lVert x - x_{b} \right\rVert^{2}+ \frac{1}{2}\left\lVert \epsilon \right\rVert^{2}$$ with $$ Hx -y = \epsilon ...
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42 views

What's a Schwartz-Bruhat Function

Let $X$ be a locally compact abelian group and $f: X \rightarrow \mathbb{C}$ a continuous map. There are several definitions of what it means for $f$ to be a Schwartz-Bruhat function. If $X = ...
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A Vietoris (hyperspace) functor for commutative unital $C^\ast$-algebras

There is a well known hyperspace functor $V \colon \mathbf{KHaus} \to \mathbf{KHaus}$ on the category of compact Hausdorff spaces. This is defined as follows: For objects $V(X) = \{K \subseteq X ...
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1answer
26 views

Factoring out the trace of a matrix

This question is related to a derivation step in " A Duality View of Spectral Methods for Dimensionality Reduction" Xiao et al. 2006 When deriving the dual equation for Maximum Variance Unfolding ...
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1answer
46 views

Gelfand duality restricted to the category of Stone spaces

It is well known that the category of unital commutative $C^\ast$-algebras is dually equivalent to the category $\mathbf{KHaus}$ of compact Hausdorff spaces. I have found that restricting the ...
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1answer
155 views

Primal feasible solution implies Dual optimal solution?

Every feasible solution of P puts an upper(or lower, depending on whether it is a maximization or minimization problem) bound on the optimal solution of D(assuming of course that D has a feasible ...