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

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Fenchel duality in conic program

The question is from the textbook Convex Optimization Algorithms, prof. Bertsekas, p.511 A special case of Fenchel duality is the following: \begin{equation} ...
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27 views

The dual function of composite functions

Given $X$ $Y$ are two finite dimensional Hilbert space. Let $K$: $X\to Y$ be linear and $F$: $Y\to \mathbb R^+$ is convex. Let us use $F^\ast$ to denote the dual (conjugate) function of $F$. Recall $$ ...
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42 views

Dual space isomorphism and the dual representation

Let $V$ be a complex finite-dimensional vector space. Then there always exists an isomorphism $V \simeq V^*$, where $V^*$ is the dual space. The isomorphism can be fixed by choosing a non-degenerate ...
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59 views

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 ...
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36 views

The space of continuous functions as a dual space

Let $X$ be some topological Hausdorff space and $C_b(X)$ the space of bounded complex continuous functions on $X$. Is there a Banach space $B$ such that $B^* \simeq C_b (X)$? I know of a very similar ...
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Question regarding characters and point open topology2

this is a follow-up question for the following one: Dual group of G with point open topology is an intersection of C(G,T) and a closed set In the book of Banaszczyk - "Additive subgroups of ...
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52 views
+50

What's wrong in this dual derivation?

I have a function in the form \begin{align} f(q,M)=\sup_{0\leq \alpha \leq 1} -\alpha^T (R\odot M)\alpha+\alpha^Tq \end{align} which is a dual of a minimization problem, where $R$ and $M$ are ...
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1answer
24 views

$\dim\ker T = \dim\ker T^{\vee}$

Let $V,W$ be vector spaces over $\mathbb{F}$, and $T:V\to W$ a linear map between the spaces. Let $T^{\vee}:W^{\vee}\to V^{\vee}$ denote the dual map of $T$ ($V^{\vee}, W^{\vee}$ are the dual ...
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34 views

Dual Core in Cooperative Game Theory

I'm a bit confused over if the dual core of a game is the same as the core of the original game. Definition of dual game: $$ v^*(S) = v(N) - v( N \setminus S ), \forall~ S \subseteq N.\, $$ I then ...
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11 views

How can we constrain lagrange multipliers in svm dual by adding constraints in primal problem?

Consider svm-dual,i.e., \begin{align} &\text{maximize} \sum_{i=1}^n \alpha_i-\frac{1}{2\lambda} \sum_{i,j=1}^n \alpha_i \alpha_j y_i y_j K(x_i,x_j)\cr &\text{subject to, } 0\leq \alpha_i ...
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27 views

What is the intuition behind Gordan's theorem?

Gordan's theorem: Exactly one of the following has a solution: $y^TA > 0$ for some $y \in \mathbb R^m$ $Ax = 0$ ;$ x \geq 0$ for some non-zero $x \in \mathbb R^n$ I am not looking for the ...
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28 views

Why L belongs to the dual space $H^{-1}$

I'm studying pde using Evans book. In chapter 6 he introduces second order partial differential operators for example : $L= \sum{D_i(a_{ij}D_j(u))+\sum{b_iD_i(u)+cu}}$ I can't understand why $L \in ...
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10 views

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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10 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
20 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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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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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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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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14 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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34 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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40 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
36 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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26 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
54 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
48 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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66 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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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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47 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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42 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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23 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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71 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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36 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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62 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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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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70 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
37 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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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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60 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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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
36 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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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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35 views

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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32 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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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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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 ...