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

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36 views

Continuous inculsion of the dual of continuous included Banach spaces

If $B$ and $C$ are Banach spaces and $B \subset C$ with the inclusion being continuous. If it true that the set of continuous linear functionals on $C$, $C'$, is continuous included in the set of ...
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82 views

Symmetric bilinear forms and (continuous) dual spaces

Let $V$ be an infinite dimensional locally compact vector space over a field $k$ (the field $k$ has the discrete topology and on $V$ we fix the linear topology ). Moreover suppose that on $V$ is ...
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29 views

Is there a non-trivial character on any locally compact Abelian group?

Let $G$ be a locally compact Abelian group. Is there a non-trivial character on $G$? indeed, i want the proof of the existence of a non-trivial character on a locally compact Abelian group.
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1answer
17 views

Nonzero linear transposition

Let $U,V$ be vector spaces and $U^\intercal, V^\intercal$ their duals. Let $T \in \operatorname{Hom}(V,U)$(A linear map) Denote by $T^\intercal \in \operatorname{Hom}(U^\intercal,V^\intercal)$ the ...
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20 views

Duality between $Ax<b$ and another system, but Gordan's just misses

I am trying to show that $$Ax < b$$ Is feasible iff $$ A^T y =0 , b^Ty + s = 0, (y,s) \ge 0, (y,s) \ne 0$$ Is infeasible. Work So Far Now when I try to hit this with Gordan's Lemma I seem ...
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1answer
51 views

Prove that the Pontryagin dual of $\mathbb{R}$ is $\mathbb{R}$.

From Wikipedia: the group of real numbers $\mathbb{R}$, is isomorphic to its own dual; the characters on $\mathbb{R}$ are of the form $r \to e^{i\theta r}$. How can I prove this assertion?
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24 views

Cartier dual of finite flat group scheme

Let $F$ be a finite flat group scheme over an algebraically closed field $k$ of characteristic 0. Note that I'm not assuming $F$ to be commutative. Does the Cartier dual $F^D$ of $F$ exist? That ...
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1answer
29 views

Dual maps and Differentiation

Let $D: F[t]_{n} \rightarrow F[t]_{n}$ be the differential map. Let $f \in (F[t]_{n})^{*}$ be the functional assigning to each $y(t) \in F[t]_{n}$ the integral $\int_{0}^{1}y(t) dt$. Compute the ...
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1answer
30 views

dual basis and existence of non-zero functional

Consider the following statements, given $V$ is a vector space. (A): Given $v \in V, v \neq 0$, there exists $f \in V'$ such that $f(v) \neq0$. (B): Given a set of linearly independent vectors ...
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3answers
79 views

Show that $T^*$ is one to one $\implies$ $T$ is onto

Let $V$ and $W$ be $F$-vector space, and $V^*$ and $W^*$ be the dual space of $V$ and $W$, respectly. Let $T:V \to W$ be a linear transformation. Define $T^*:W^* \to V^*$ by $T^*(f)=f\circ T$ for all ...
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21 views

knapsack problem using duality

I have this knapsack problem \begin{align} \max {z}=10&x_1+8x_2-13x_3 +10x_4+10x_5 + 5x_6\\[0.4cm] \text{s.t. }\quad\,\,\,7&x_1+6x_2+10x_3+\,\,\,8x_4 +\,\,\,9x_5 + 5x_6\le39 \\[0.2cm]& ...
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28 views

Proving that$ B^*$ is a dual basis

Suppose we have: $$ E= \mathbb{R^2}[t]=\{p(t)=a_0+a_1t+a_2t^2\} $$ $$m_0, m_1, m_2$$ Where $m_0, m_1, m_2$ are different real numbers. We define $$F_i:E\rightarrow \mathbb{R}, i = 0,1,2$$ By ...
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1answer
23 views

How is $v^T(Ax-b) = -b^Tv+(A^Tv)^Tx$?

This might seem a silly question but I am thoroughly confused. Given $c^Tx+v^T(Ax-b) = -b^Tv+(c+A^Tv)^Tx$ I get that it can be written as the second term in the RHS, but what about the first one ...
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1answer
56 views

linear transformation $T$ is onto if and only if $T^*$ is one to one

Let $V$ and $W$ be $F$-vector space, and $V^*$ and $W^*$ be the dual space of $V$ and $W$, respectly. Let $T:V \to W$ be a linear transformation. Define $T^*:W^* \to V^*$ by $T^*(f)=f\circ T$ for all ...
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14 views

SVM: How to normalize |WX| > 0 into |WX| = 1

Question What are the reason/basis/rationale and the actual steps and design/mechanism behind to do the normalization to convert |WX| > 0 into |WX| = 1 in the process of getting the optimal W for the ...
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36 views

How is the dual of a LPP defined?

The dual of a cone is the set of vectors which have a non-positive dot product with any vector in the original cone.This definition does not seem to be valid in the LPP formulation wherein the ...
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29 views

Why is one of the following always true in a given matrix A?

Consider A, a given matrix. I want to show that exactly one of the following holds: 1) $\exists x\neq 0$ such that $Ax=0$ 2) $\exists p$ such that $p^TA>0$ I tried proving that this is ...
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12 views

Dual of the following non-linear program

I am new to optimization and understanding some concept. I understood how duality work and tried applying it some linear programs. I followed the same for non-linear programs but I end up wth a ...
2
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1answer
41 views

Dual map is zero if and only if map is zero

A problem from Linear Algebra Done Right (Third Ed): Suppose $W$ is finite dimensional and $T \in \mathcal{L}(V, W)$. Prove that $T=0$ if and only if its dual $T' \in \mathcal{L}(W', V')=0$. I am ...
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25 views

Can the non-uniqueness of a linear program's dual feasible set be exploited?

I was originally under the impression that a primal LP had a single corresponding dual feasible set. However, it is possible to alter the primal to an algebraically equivalent form which has a ...
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2answers
107 views

Is $C[0,1]$ reflexive?

I.e. is the embedding $C[0,1]\hookrightarrow \left( C[0,1] \right)^{**}$ surjective? I am having a hard time answering that question. It would be enough to find a closed subspace of $C[0,1]$ which ...
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0answers
36 views

Primal/Dual Simplex methods clarification

I have several questions regarding these methods. Primal Simplex Method Does the pivot element always have to be a positive entry in the table? Does the RHS always have to be positive in the pivot ...
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21 views

Does Slater's condition for both primal and dual imply compactness of dual solution set?

Consider a convex optimization problem (P) and its dual problem (D). If the solution set for (P) is compact and Slater's condition holds for both (P) and (D). Is the solution set for (D) compact? My ...
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1answer
27 views

problem on annihilators on finite dimensional spaces

Suppose $V$ and $W$ are subspaces of a finite-dimensional vector space $U$. Show that if $V^0 \subset W^0$ then $W \subset V$ This is an exercise problem in Linear Algebra Done Right, 3rd ...
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2answers
90 views

Is there any comprehensive definition of the opposite category $\mathcal C^{op}$ of a category $\mathcal C$?

In the definition of the opposite category $\mathcal C^{op}$ of a category $\mathcal C$, it is said that : objects remain the same and arrows' directions are changed (that is, and arrow $f:A \to B$ in ...
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14 views

Explain the dual problem to D-optimal design problem

Given the following D-optimal design problem $$ \text{minimize } \log \det (\sum_{i=1}^p x_i v_i v_i^T)^{-1}\\ \text{subject to } x \geq 0, {\bf{1}}^T x = 1 $$ Find the dual problem. I don't ...
4
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1answer
146 views

Gradient of the dual function for a nonlinear program

I'm attempting to find a proof for a property from Floudas' Nonlinear and Mixed-Integer Optimization book. Consider a nonlinear optimization problem of the form \begin{align} \min_{{\bf x}}&\quad ...
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45 views

Duality for modules

I'm doing exercises given in Maclane "Homology", and I have problems with the following exercise. The task is to state the dual of this proposition: "For submodules $S_t\subset B, \ t\in T$ the ...
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1answer
35 views

Properties about reflexive space

I'm studying fuctional analysis and specifically reflexive spaces. My textbook has a introductory level, so don't cover so many things. My questions are: 1) If $X$ and $Y$ are isomorphics and $X$ is ...
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1answer
71 views

Minimum infinity norm control problem

I am having trouble understanding Example 2 of section 5.9 of Luenberger's Optimization by Vector Space Methods. The problem is to select a current $u(t)$ on $[0,1]$ to drive a motor governed by $$ ...
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1answer
59 views

Dual program is wrong. Authors claim is right.

In a well respected book, I found the following. The authors claim that it is correct. But I think it is wrong. This is the primal Linear Problem: $$ \begin{array}{cccc} ...
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2answers
25 views

functions acting as linear functionals on their dual space

Supposing $f\in L^p$, where p and q are conjugate exponents, what does it mean that "f is completely determined by its action as a linear functional on $L^q$"? (Quoting Folland's Real Analysis here). ...
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33 views

Cesaro and Tandori sequence spaces, representations and duality

Definitions. Fix $1\leq p\leq\infty$. Given a scalar sequence $a=(a_n)_{n=1}^\infty$, denote by $\tilde{a}=(\tilde{a}_n)_{n=1}^\infty$, where each $\tilde{a}_n=\sup_{k\geq n}|a_k|$. Now we define ...
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1answer
47 views

Equivalent condition for a linear map to coincide with another restricted to a line modulo certain hyperplane.

Let $p$ be a line of the vector space $K^{n+1}$, and let $H$ be a hyperplane of $K^{n+1}$ such that $p\subseteq H$. We may also interpretate $H$ as a line $q\subseteq(K^{n+1})^{*}$. Let ...
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3answers
136 views

Concerning $f(x_1, \dots , x_n)$

I am not getting even an intuition as how to do this problem. Please help me with a solution.. Let $n$ be a positive integer and $F$ a field. Let $W$ be the set of all vectors $(x_1, \dots , x_n)$ in ...
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28 views

Visualising the dual function

Consider the linear programming problem: min $-x_1-2x_2$ s.t. $3x_1+2x_2-6\leq 0$ $ -x_1+2x_2 -4\leq 0$ $0 \leq x_1 \leq 3/2, 0 \leq x_2 \leq 3/2$ Find the Lagrange dual objective function ...
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57 views

Proof that there exists a polynomial $q$ such that for all polynomials $p$ we have $\int_{-1}^1p(x)q(x)dx=p(2)$ [closed]

Let $V$ be the real vector space of polynomials of degree $\leq2$ and the inner product is $\langle p,q\rangle =\int_{-1}^1p(x)q(x)dx$. How do I show that there exists a $q\in V$ such that for all ...
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3answers
51 views

Proof that there exist constants $a_i$ such that $\int_0^1 f(x)e^xdx=\sum_{i=1}^na_if(i)$ for polynomial $f(x)$ of degree less than $n$

How do I show that for positive integer $n$ and $f(x)$ all real polynomial functions of degree less than $n$ there exist constants $a_i$ such that $$\int_0^1 f(x)e^xdx=\sum_{i=1}^na_if(i)?$$ I thought ...
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An unfamiliar constraint in dual of LP for 'extreme point' optimal solution

I am trying to understand two-stage chance-constrained LP and reading a paper, which has a title "A branch-and-cut decomposition algorithm for solving chance-constrained mathematical programs with ...
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29 views

Linear programming duality problem

I need to solve this problem linear programming task using dual graphical solution. Task I tried creating it's dual form using those rules: Rules And I got my dual task as follows: \begin{align} ...
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1answer
48 views

What is the relationship between a quotient space and annihalator?

If we have a vector space $V$ and subspace $W$, we have that $$\dim(V/W) = \dim V - \dim W.$$ Similarly for the annihilator $W^{\circ}$ we have that $$\dim W^{\circ} = \dim V - \dim W.$$ What is ...
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1answer
49 views

First homology group of non-orientable manifold

Can I use Poincaré duality to prove that the first homology group of a non-orientable manifold $M$ is not zero? I only need to prove that $H_1(M;\mathbb Z_2)\ne0$, and by the universal coefficient ...
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27 views

Prove that the dual space of $l^1$ is $l^\infty$.

Prove that the dual space of $l^1$ is $l^\infty$. In order to show this I have to show that $l^{1^{'}}\cong l^\infty$.I have to find a bounded linear operator from $LHS$ to $RHS$ I take $f\in ...
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2answers
41 views

Proof involving weak and weak-* convergence

"Prove that if $(f_n)_{n=1}^\infty\subset X'$ converges strongly in $X'$ and a sequence, $(x_n)_{n=1}^\infty\subset X$ converges weakly in $X$, then, $f_n(x_n)\to f(x),\,n\to\infty$." My attempt: ...
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35 views

Separation in Boolean algebras

I am looking for a separation-like result for Boolean algebras which is intuitively clear to me. Suppose that $B$ is a Boolean algebra whose set of positive elements $B^+$ does not have countable ...
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25 views

Conic formulation. Finding a point minimizing the maximum distance to a set of points.

I have to formulate (and I don't know) as a conic problem the next: Problem: Given a set of points $D=\{a_1,a_2, \dots, a_n\} \subset R^2$. Write like a conic problem the problem to find a point ...
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1answer
40 views

compact oriented $4k$-manifold-> euler characteristic is congruent to the signature mod $2$

Let $M$ be a compact oriented $4k$-manifold, $\chi_M$ the euler characteristic of $M$ and $sig(M)$ the signature of $M$. Why is $\chi_M \equiv sig(M)$ mod $2$? In the book " a concise course in ...
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51 views

Is this method of finding a “dual curve” correct?

I have a very limited exposure to projective geometry, but I'm having fun exploring the concept of duality. In particular I'd like to know if this naive method of finding a "dual" curve to a given ...
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21 views

If X* is separable X is also separable.X is normed vector space

I know the proof of this fact by contradiction. Is there any proof without contradiction or the reason why this happen? The Banach space $L(X,\mathbb{R})$ is called the norm dual of $X$ and is denoted ...
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2answers
31 views

Proving the transpose / dual map is well defined.

The definition for a dual map is as follows: The dual map, or transpose of linear $f:V \rightarrow W $ is given by $f^t(g)(v) = g(f(v)) $ for $\forall g \in W^* , v \in V $. In my lecture ...