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

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45 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
30 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
22 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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25 views

when will dual optimal solution and primal optimal solution will be equal? [closed]

I don't mean like the optimizing value of the primal and dual what I mean is the individual feasible solutions of primal and dual being equal.An example would be good.
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1answer
28 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
32 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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34 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
26 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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16 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
25 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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49 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
18 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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0answers
44 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
52 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
68 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
24 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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31 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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15 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
20 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
32 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
50 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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27 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
41 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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0answers
32 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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27 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
79 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
28 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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0answers
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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21 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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33 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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54 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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23 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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43 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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59 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 ...
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28 views

If the supremum is finite, then the value is attained.

In a linear programming proof we have the: $\sup\{c^Tx: Ax \le b\}$ This supremem can be $\infty$, or defined as $-\infty$, if there are no vectors x such that $Ax \le b$. But it is stated that ...
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1answer
47 views

How to show these two problems have equivalent solutions

I have two problems, where $A$ is positive definite: $$\inf \{ x^TAx + b^Tx : 1-x^Tx \ge 0,\ x \in \mathbb R^n\} \ (1)$$ and $$ max_\lambda \ q(\lambda) = -0.25b^T(A+\lambda I)^{-1}b - \lambda : ...
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54 views

Sequential compactness in weak topology

When the Banach space $V^*$ is reflexive, we have the unit ball in $V^*$ is weak$^*$ sequentially compact. For a Banach space $V^*$ that might not be reflexive, we have to assume that $V$ ...
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2answers
89 views

A proof for $\widehat{\Bbb Z_{p^\infty}}\cong Z_p$

According to wikipedia, the Pontryagin dual of a Prüfer group is isomorphic to a group of p-adic integers. Where can I find a proof for it on the internet?
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53 views

Can we show it without involving that $V=V^{**}$ are canonically isomorph?

My text proves the following Theorem. Let $V$ be a vector space over $F$ and $B=\{ v_1, \ldots , v_n \}$ a basis of $V$. Then there is exactly one basis $B^*=\{ f_1, \ldots , f_n \}$ of $V^*$ with ...
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2answers
66 views

Duality in quadratically constrained quadratic program

I have been given the primal quadratic program with a single quadratic constraint as given below: $$ \text{min} ~~~~~~~~~~~~~~~~~~~~~~~~~ \frac{1}{2}x^{T}Qx $$ \begin{align*} \text{subject ...
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1answer
39 views

If $X\subset Y$ then $X^*\subset Y^*$

Is the following true, If $X$ and $Y$ are Banach spaces and $X\subset Y$, then $X^*\subset Y^*$. One argument for this is the following let $i:X\to Y$ be the identity map which implies its one to ...
2
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1answer
134 views

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 ...
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62 views

Understanding dual spaces

My professor asked me to give him an example about dual spaces from our real life so I was thinking if we take the set of all our actions as the space X can the space of all our thoughts be the dual ...
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1answer
55 views

Connected Pontryagin dual

The dual group to a compact abelian group is discrete so in particular very much disconnected. I was trying to invent an example of a connected locally compact abelian group with connected dual which ...
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1answer
30 views

Show that $E^*\neq \{0\}$ iff $E\neq \{0\}$ [closed]

Let $E$ be a vectorspace and $E^*$ be a algebraic dual of $E$. Show that $E^*\neq \{0\}$ iff $E\neq \{0\}$