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

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23 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
26 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
40 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
51 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
23 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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42 views

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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2answers
59 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
45 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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67 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
27 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
43 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
26 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
28 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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0answers
34 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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1answer
29 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
27 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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58 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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0answers
21 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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46 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
80 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 ...
3
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1answer
80 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
27 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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0answers
35 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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18 views

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
25 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
37 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
85 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 ...
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29 views

Examples of non-self-dual Hilbert spaces?

I'm looking for some basic examples of non-self-dual Hilbert spaces, as well as basic examples of self-dual complex Hilbert spaces. Concrete examples would be helpful.
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1answer
22 views

Solving a linear program using just one call to a procedure that gives a feasible solution.

Suppose we have some procedure $F$ which takes any set of linear constraints and either returns either infeasible or returns a vector satisfying these constraints. If we now take a linear program ...
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1answer
28 views

Dual space of weighted $L^p(\omega)$

Let $\omega \in A_p$, where $A_p$ is the family of Muckenhoupt weights. I'm wondering what is the topological dual space of $L^p(\omega)$. Is it isometrically isomorphic to $L^q(\omega)$? (1/p + 1/q = ...
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1answer
50 views

dual of $L^\infty$ or $l^{\infty}$ spaces and characters of these spaces

Let $\nu$ be a $\sigma$-additive probability measure on some standard Borel space $(X,\Sigma)$. By Gelfand's transform or by Stone-Cech compactification $L^{\infty}(X,\nu)$ is isomorphic to ...
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33 views

Legendre Transform of this function

Is the Legendre-Fenchel transform of $$f(x)=1-\sqrt{1-|x|^2}, x\in B(0,1)\subset\mathbb{R}^n$$ just $$f^*(x^*)=-1+\sqrt{1+|x^*|^2}?$$ I calculated this using the table here ...
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29 views

Dual formulation of weak $L^p$

Let $1<p\leq \infty$. Then we have$$||f||_{L^{p, \infty}(X,d\mu)} \sim_psup\{\mu(E)^\frac{-1}{p'}|\int_E f d\mu|:0<\mu(E)<\infty\}$$ Where$||f||_{L^{p, \infty}(X,d\mu)} = ...
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1answer
83 views

Show both relaxations of boolean LP give equal lower bounds

Given the boolean LP: $$\text{Minimize}\;\; c^Tx$$ $$\text{Subject to}\;\; Ax \leq b$$ $$\hspace{57mm} x_i(1-x_i)=0\;\; i=1,...,n$$ Show that the LP relaxation: $$\text{Minimize}\;\; c^Tx$$ ...
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1answer
16 views

Dual cone of $K = \{(x,t) \mid \| \boldsymbol{x} \|_1 \le t \}$

This slide shows the dual cone of $K = \{(x,t) \mid \| \boldsymbol{x} \|_1 \le t \}$ is $K^{*} = \{(x,t) \mid \| \boldsymbol{x} \|_{\infty} \le t\}$. Is it right? How is it proved?
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1answer
32 views

Primal of Dual of LP problem

Given that the following relation holds: $$\begin{align*} &\textbf{Primal problem} \\ &\max Z = c^Tx \\ &s.t. \\ &Ax \leq b \\ & x \geq 0\end{align*}$$ $\Longrightarrow$ ...
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23 views

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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22 views

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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37 views

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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57 views

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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24 views

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. } & ...
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44 views

Is the dual of this semi-definite problem derived correctly?

I have a very peculiar problem. I have a semi-definite problem. My problem is \begin{align} \min_{t,A}~&t\\ &L>=0\\ &A>=0\\ &A.*M=Y_M\\ &A.*I=I\\ ...
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62 views

Alternate proof of a result on dual spaces: what is wrong with it?

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. ...
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31 views

Why is this conic dual problem infeasible?

The problem is: $$\min \ x_2 : Ax -b = [x_1 \ 2x_2 \ x_1]^T \ge_{L^3} 0$$ where $L^m$ is the Lorentz cone. Which I found to have an optimal solution when $x_2 = 0$. I have shown that the conic ...