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

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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
13 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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77 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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6 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 ...
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1answer
73 views
+50

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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41 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
26 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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44 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
53 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
22 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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26 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
45 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
129 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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1answer
25 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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3answers
56 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
47 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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0answers
12 views

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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0answers
23 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} ...
3
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1answer
44 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
23 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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24 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
40 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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21 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
27 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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14 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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20 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
23 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 ...
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31 views

Confusion about Theorem structure of Structure theorem for Gaussian measures

Structure theorem for Gaussian measures is explained in the book by Fritz P. & D. Victoir "Multidimensional Stochastic process as a rough paths" on page 606. I have some confusion as illustrated ...
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13 views

Formualte the Lagrange dual $(D _1 )$ of $ (P _1 )$,

(a) One way to formulate the projection problem is $$\min \frac{1}{2}\|x-a\|^2$$ $$s.t \> Ax=0$$ where $A$ is a matrix such that $L = \{x : Ax = 0\}$ (i) Formulate the Lagrange dual $(D _1 )$ of ...
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1answer
14 views

Prove constrained problem has no duality gap

For an exam I have to prove that if a constrained problem has no duality gap for some $(l,g)$ and $x$, then $x$ is a global minimum point for the constrained problem. Do you think an example is ...
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28 views

Definition of dual connection in riemannian geometry

If D is a connection on a vector bundle E, we define the dual connection D* so that $$d(v^*,w)=(D^*v^*)(w)+v^*(Dw)$$ I understand why this seems the natural thing to do. Why is the following not ...
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23 views

Conditions for a system of linear forms to be a basis

Let $V$ be the vector space of all polynomial functions from $\Bbb R [x] _{\le 3}$ to $\Bbb R [x] _{\le 3}$. Consider the linear forms $f_i$ defined for $p$ in $V$, as $f_i(p)=p(a_i)$, where $a_i \in ...
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50 views

existence of a linear functional

I am trying to solve this exercise: Let $\ell_\infty$ be the set of real infinite bounded sequences. Prove that there exists $\lambda \in (\ell_\infty)^*$ such that for all ...
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39 views

Relation between annihilator of kernel and range of the transpose

Let $A:X\to Y$ be a continuous linear map. Let $A^*:Y^*\to X^*$ denote the transpose of $A$ (also referred to as the dual transformation of $A$) Let $\ker A ^⊥$ denote the annihilator of ...
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1answer
29 views

Primal to Dual Linear Programming

I'm learning how to convert primal LP problems to dual, but not sure if I'm doing it correclty. primal: $$ \begin{align} maximize: \ \ \ \quad x_1 + 2x_2\qquad\quad \ \ \\ subject\ to:\ -2x_1 + x_2 + ...
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1answer
9 views

How to get $v ' (x + y) + w ' (−x + y) = 0$ from the dual complementary condition

I'm perusing this paper: ftp://ftp.cs.wisc.edu/pub/dmi/tech-reports/13-01.pdf and I'm stuck in how to deduce this equation: $v'(x + y) + w'(−x + y) = 0$ from this optimization problem: ...
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2answers
37 views

Linear functions and intersections of null subspaces

Let $V$ be a vector space of a finite dimension $n$ over the field $K$. Let $\phi, \psi$ be two non-zero functionals on $V$. Assume that there is no non-zero element $c \in K$ such that $\psi= c ...
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0answers
23 views

Use Fenchel Duality to minimize cTx, subject to x ∈ A∩C

minimize cTx subject to x ∈ A∩C, where $x,c∈R^n$, C is a convex closed nonempty set in $R^n$, A=a+S is an affine set, where $a∈R^n$ and S is a subspace of $R^n$, and A ∩ ri(C)≠ ∅. Use the Fenchel ...
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2answers
46 views

Minimize $x^2+y^2$, subject to… (optimal points, KKT conditions, dual theories)

I am new to this. I am self learning to get ahead of my next years course and came across this question. I thought it would be a good question to look at due to it touching an many different aspects ...
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1answer
47 views

How to solve this operation research problem using dual simplex method?

Maximize $$ z = 2x_1 -x_2 +x_3$$ Subject to constraints $$2x_1 + 3x_2 -5x_3 \ge 4$$ $$-x_1 +9x_2 -x_3 \ge 3$$ $$4x_1 ...
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1answer
82 views

Image of dual map is annihilator of kernel

Suppose $T:V\to W$ and that $V$ is finite-dimensional. I want to prove that $$\text{Im }T'=(\ker T)^0$$ where $T'$ is the dual/transpose map and $(\ker T)^0$ is the annihilator of the kernel. I know ...
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1answer
38 views

$P\cong P^\ast$ iff $P$ is a f.g projective module?

Is it true that for a noncommutative $R$, a module $P$ is f.g projective iff $\mathsf{hom}(P,R)=P^\ast \cong P$? Here's what I thought of as a proof: Since $(-)^\ast$ is additive, it preserves ...
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1answer
58 views

Trying to understand the slick proof about the dual space

In this famous MO question, a beautiful proof is given of the fact $V\cong V^\ast\iff V$ is finite dimensional. I'm trying to go through it and I'm having some trouble. First of all, I know the in ...
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1answer
53 views

Linear Programming Duality Proof

I have really no idea where to go in this problem. This is from Bertsimas Introduction to Linear Optimization, Exercise 4.26. My teacher would like us to create a primal and dual LP to solve the ...
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1answer
64 views

The image of a dual transformation is equal to the set of annihilators of the kernel of the original transformation.

I'm presented with the following question: Suppose $T:V \to W$ is a linear map and $V$ is finite dimensional. Prove Im $T'$ = (Ker $T$)$^0$. I have showed that Im $T'$ $\subseteq$ (Ker $T$)$^0$: ...
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53 views

Duality between tangent and cotangent bundles

Given a smooth manifold $M$, the cotangent bundle $T^*M$ is dual to the tangent bundle $TM$ "fiberwise", i.e. $\forall x\in M$, $T^*_x(M)=(T_x(M))^*$. Now, if the manifold is a vector space, then the ...
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31 views

Can we show that $(E\times \mathbb R)^*=E^* \times \mathbb R$ where $E$ is a Banach space?

Can we show $(E \oplus \mathbb R)^* \cong E^* \oplus \mathbb R$, where $E$ is a Banach space and $E^*$ is the dual space of $E$? What if $E$ is just a normed space or even a topological space? To be ...
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1answer
85 views

$c_0$, the space of sequences converging $0$ is complete with dual $\ell^1(\mathbb{N})$

Let $c_0$ be the space of all complex sequences $(a_n)$ such that $$\lim_{n \to \infty} |a_n| =0$$ with norm $\|(a_n)\|_{c_0} = \sup_{n} |a_n|$. Is it fair to say that: Let $\{(a_n)\}_{n \in ...
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34 views

exercise: dual of an Lp-space

I have this exercise: Let $(\Omega,\mathcal{A},\mu)$ be an arbitrary measure space and $1 <p <\infty$. Show that if $l \in \mathcal{L}^p(\mu)^*$, then there exists a sequence ...