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

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1answer
87 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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0answers
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 ...
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1answer
30 views

A closed Subspace of a reflexive Banach space is reflexive

Let $X$ be a reflexive Banach Space. Let $Y$ be a closed subspace of it.I need to show that $Y$ is reflexive as well. So as usual I consider the inclusion map $$J: Y \to Y'', J(y)=j_{y}, ...
4
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1answer
55 views

Trace-class, Hilbert Schmidt operators, $L^p(H)$: duality theorems

Let $H$ be a Hilbert space, separable if necessary, and let $tr$ be the usual trace on $L^1(H)$. It is classical theory that $K(H)^*=L^1(H)$, and $L^1(H)^*=B(H)$, via the canonical application ...
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0answers
41 views

If primal and the dual problems have feasible solutions, then both have optimal solutions

Prove that if both the primal and the dual problems have feasible solutions, then both have optimal solutions, and the optimal objective values of the two problems are equal. My attempt: Let ...
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0answers
15 views

How to draw the dual of a quadrangle determined by four points?

Draw a quadrangle determined by the points $P, Q, R, S$ and name the sides and the three diagonal points $A, B,$ and $C$. Now draw the dual of this quadragle and name its sides and diagonals $a, b$, ...
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2answers
239 views

What does “dual” mean exactly in mathematics?

I'm not a math expert but I know a little bit of calculus and theorems. I've heard things like "this result is "dual"", or this "...
0
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1answer
28 views

Solving a PL using complementary slackness conditions - dual

I have to find the optimal solution of the dual with the complementary slackness conditions. This is the primal: $\max \space\space z= x_1 - 2x_2 $ $\text{s.t.}\space\space\space\space\space ...
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0answers
14 views

Construct a Primal-Dual Linear Programming Pair such that the feasible domains of both are non-empty and bounded

I've been studying duality and I've been trying to construct primal-dual pairs that satisfy specific properties. I'm wondering if they can both have bounded feasible domains?
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0answers
66 views

Basis for the dual space of polynomials

I am stuck on the following question: Let P(n) be the space of real polynomials of degree at most n. For r ∈ ℝ define πᵣ ∈ P(n) * by πᵣ(p) = p(r). Show that π₀, π₁, ... form a basis for P(n) *, and ...
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0answers
21 views

Why does the dual problem of the SDP become a maximum eigenvalue problem?

This SDP problem with the variable $X \in \mbox{S}^n$ where $\rho \gt 0$ $$\max \mbox{Tr}(AX) - \rho \mbox{1}\lvert X \rvert \mbox{1} \\ \mathrm{subject\; to}\;\mbox{Tr}(X)=1, \\ X \succeq 0 $$ ...
2
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1answer
63 views

Prove that $\ker f=\ker g$ implies $f=cg$.

Let $V$ be a vector space of dimension $n$; let $f: V \to \Bbb K$, $g :V \to \Bbb K$ be linear maps, so that $f,g \in V^*$. Prove that if $$\ker f=\ker g,$$ then there is some $c \in \Bbb{K}$ such ...
3
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0answers
64 views

Explicit computation of Serre duality

Given a projective non-singular curve $X$, the a Serre duality asserts an isomorphism between $H^0(X,\Omega^1_X)$ and dual of $H^1(X,\mathcal{O}_X)$. My question is how to compute the dual elements in ...
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0answers
43 views

When the size of minimum edge cover is equal to the size of maximum independent set?

We know that the size of the minimum edge cover is always equal to or greater than the size of the maximum independent set of the same graph. I want to know is there any special type of graphs that ...
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0answers
108 views

Proof that $\text{im}(g)^\top=\ker(f)^\top$ if $\text{im}(f)=\ker(g)$

Let $f:V\rightarrow W$ and $g:W\rightarrow X$ be two linear maps with $\text{im}(f)=\ker(g)$. How do I prove that $\text{im}(g)^\top=\ker(f)^\top$? I am allowed to use the fact that if $f$ is ...
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1answer
86 views

Proof that $\text{im} f^\top=\ker j_U^\top$

Let $f:V\rightarrow W$ be a surjective linear map with $U=\ker(f)\subset V$. Let $j_U:U\hookrightarrow V$ be the inclusion of $U$ in $V$. How do I show that: $\text{im} f^\top=\ker j_U^\top$? I ...
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1answer
97 views

Dual space and covectors: force, work and energy [closed]

I am an engineer trying to understand the mathematical definitions and physical significance of vectors, dual vectors, and dual spaces. I understand how we take dot products and the covector "eats" ...
2
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1answer
43 views

The cycle space of a planar graph is the cut space of its dual graph

I am trying to understand the following statement on wikipedia: The cycle space of a planar graph is the cut space of its dual graph, and vice versa. Suppose we have a cycle space $\mathcal ...
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0answers
52 views

Image of a dual map is equal to Image$^{\perp\perp}$

Given two dual vector spaces $V$ and $V^*$ as well as linear maps $\phi: V \to V$ and $\phi^*: V^* \to V^*$ such that $\phi$ and $\phi^*$ are dual, i.e. there exists a non-degenerate bilinear form ...
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1answer
29 views

Duality of a linear programming problem in matrix form?

trying to find the duality of the LP problem in matrix/vector form: min c1Tx1 + c2Tx2 s.t. A1x1 + A2x2 = b x1>= 0 I get that the duality of like min cTx s.t. Ax = b x>=0 would be max ...
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1answer
56 views

Derive this variant of Farkas' lemma, through another variant of Farkas' lemma.

Derive the following variant of Farkas' lemma: For each $mxn$ matrix A and vector $b\in\mathbb{R^m}$ one of the following statements is true: $\exists x\in\mathbb{R^n}$ such that $Ax=b$ $\exists ...
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1answer
75 views

Vector space of functions $\mathbb{R}\rightarrow\mathbb{R}$

Let $V$ be the vector space of functions $\mathbb{R}\rightarrow\mathbb{R}$ and $W$ the subspace generated by the functions $f_1=\cos(x),f_2=\sin(x),f_3=\sin(2x)$. For $k=1,2,3$ let $\phi_k\in W^*$ ...
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0answers
16 views

Bijection between left integrals and right orthogonal associative bilinear forms in a bialgebra

Suppose $H$ is a bialgebra over a PID $R$. I am trying to understand the bijection between right orthogonal associative bilinear forms on $H^*$ and left integrals of $H$. Specifically, I have trouble ...
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1answer
56 views

How to show the quotient of dual space and the annihilator of its subspace is equal to dual of the subspace? [duplicate]

Let $W$ be a subspace of a vector space $V$ over a field $F$. Let $i : W \to V $ denote the inclusion map. Show that $ \pi: V^*\to W^*$given by $\pi(f) = f\circ i$ is a surjective linear map, with ...
0
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1answer
32 views

Let $U \le V^{\ast}$. If $f(u) = 0$ for all $f \in U$ implies $u = 0$, then $U = V^{\ast}$.

Let $V$ be a finite-dimensional vector space over $K$, and let $V^* := \{ f : V \to K : \mbox{ f is linear } \}$ be the space of linear functionals, the so called dual space of $V$. Also let $U \le ...
1
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1answer
61 views

Conormal Variety and fiber of a projection

Let be $X$ a projective irreducible variety in $\mathbb{P}^{N}$ of dimension $n$. Let be $ V_0 = \{(p,H)\in\mathbb{P}^N\times (\mathbb{P}^{N})^{*} \mid p\in X_{sm},\ T_p X \subset H\}$ and $V$ its ...
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0answers
25 views

identify $u(n)$ with $u(n)^*$

I'm checking right now a book, in which the authors say, that they identify the dual spaces of the lie algeba $u(n) = \{A \in \mathbb{C}^{n \times n} | \overline{A} = -A^T \}$ again with $u(n)$ by ...
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0answers
23 views

Duality in Langrange Multiplier

The problem is to minimize $$f(x) = x^T.x$$ subject to condition $$Ax = b$$ With the help of Lagrange Multipliers, which gives the equation $$L(x,\lambda) = x^Tx + {\lambda}^T(Ax-b)$$ The solution ...
2
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1answer
37 views

If a subspace of $X^*$ is weak*-dense, does it separate points?

Here $X$ is some normed space. I know the converse is true, but I don't know a proof for the other direction. That is, if $F\subset X^*$ is a subspace that is weak*-dense how would one show that ...
4
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2answers
39 views

Showing two norms on $\mathbb{R}^n$ are dual

I am having trouble showing the following result. If $A$ is a positive definite matrix, then the norms (on $\mathbb{R}^n$) $\|x\|_A:= \sqrt{x^\top A x}$ and $\|y\|_{A^{-1}}:= \sqrt{y^\top A^{-1} ...
0
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1answer
48 views

why this vector space is not isomorphic to its double-dual [duplicate]

let $V$ be the vector space of all sequences of real numbers which are eventually $0$, that is $ V=\{s=(a_1,a_2, . . .) | \exists N$ such that $a_N = a_{N+1} = · · · = 0\}$. We know the natural ...
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2answers
67 views

Show that the dual space of a subspace of $V$ can be identified with $V$

Let $V$ be the vector space of all sequences of real numbers $ V=\{s=(a_1, a_2, . . .) | a_i ∈ \mathbb{R} \}$. Also let $U$ be the vector space of all sequences of real numbers which are eventually ...
2
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0answers
37 views

Fenchel duality of infinity norm

The minimization problem is $\min\limits_{f_i} \sum^K_{i=1} \|f_i(\mathbf{p})\|_\infty$ Could someone explain how the Fenchel duality is used so the primal-dual formation becomes $$\min_{f_i} ...
4
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1answer
59 views

Definition of annihilator is not clear

I have a question regarding the following definition of an annihilator of a finite dimensional vector space. I think I understand the two definitions but I don't really get the link implied by the ...
3
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0answers
66 views

“Strange” notion of naturality of the cap product in the proof of Alexander-Lefschetz-Poincaré Duality

I was trying to understand the proof (the introduction to the proof actually) made in Bredon's book when I encountered a problem in this passage (Page 349) Suppose that $(K,L)\subset(U,V)$ as ...
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1answer
54 views

What can we say about the dual of an $R$-module homomorphism?

Suppose $R$ is some ring (not necessarily commutative) and let $M,\,N$ be $R$-modules. Now let $f:M\to N$ be an $R$-module homomorphism. If ${}^\ast$ denotes duality, then we can also consider the ...
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1answer
39 views

What are the implications of the dual norm of a norm?

I was doing a series of questions proving that the dual norm of $l_p$ is $l_q$, where $p,q$ satisfies $\frac{1}{p} + \frac{1}{q} = 1$. I was able to prove this result but I do not see the point of ...
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0answers
19 views

SVM why to switch to dual Lagrange?

I am newbie reading Support Vector Machines Explained. Translation to dual Quadratic program (p.3 - p.4) in SVM explaining article I mentioned confused me because I understand basic Lagrange ...
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3answers
65 views

Divergence theorem in $H^1(\Omega)$.

Let $u,v\in H^1(\Omega)$, where $\Omega$ is a Lipschitz domain in $\Bbb R^n$. It is my understanding that the divergence theorem tells us $$\int_\Omega\nabla u\cdot\nabla ...
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1answer
53 views

Proof for linearity on tensor products

Theorem: Let $U$ and $V$ be vector spaces. Let $\mathbf{u}^* \in U^*$. Define $\mathbf{f} : U \otimes V \to V$: $$\mathbf{f}\left(\sum_{r} \mathbf{u}_r \otimes \mathbf{v}_r\right) = \sum_{r} ...
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0answers
21 views

The dual of the space of $p$-locally integrable functions

If $X$ is a space of finite measure, what is the dual space of $L^p _{loc}$ (the space of locally $p$-integrable functions)? When $p=1$, a good answer has already been provided. What is known for $p ...
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2answers
274 views

What is the interest of duality in algebra, and in general in mathematics?

Before to ask my question I precise I'm a chemist, I ask this question because it makes me crazy to don't understand something I learnt in school. So I had two years ago a small chapter about ...
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1answer
54 views

Fenchel Duality in Prof. Bertsekas' lecture

Please see this link, p.39-41 (sufficient to answer my question), before (1.47) for detailed. For convenience, the relevant part is shown as: I am confused in two things: The ...
4
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0answers
136 views

When is the Lagrangian dual function smooth?

Consider a nonlinear optimization problem of the form \begin{align} \min_{x}&\quad f(x)\\ \nonumber \text{subject to } \quad&h_i(x) = 0,\,i=1,\ldots,I\\ \nonumber \quad&g_j(x) \le ...
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1answer
47 views

Defining the dual-comodule of a comodule?

As is well known, every left module has a dual, which is a right module. How does this work for comodules? More explicitly, does there exist a notion of the ...
2
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1answer
68 views

Conjugacy relation in the primal and dual problem

The following is my derivation in the Conjugacy relation in the primal and dual problem. I am shaky in it; so hope for some advices. Consider the following problem, $f(x),g(x)$ are convex ...
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2answers
36 views

Reference Request: Soft handed text on duality theory?

Can anyone recommend a text on duality theory which includes basic formulation of the primal and dual formulation and some introduction to minimax problems? Preferably having some computation in ...
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2answers
55 views

Dimension of a null-space (Halmos)

I am working on problem 3 from the exercises following the section on Annihilators in the text "Finite Dimensional Vector Spaces". Problem: Prove that if $y$ is a linear functional on an ...
5
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1answer
141 views

Duality theory and nonlinear optimization

I have been studying nonlinear optimization recently and have come across some results that I need clarification for. I will do my best to explain them in detail below, providing citations where ...
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0answers
35 views

Does a zero duality gap imply global optimality?

Let's say we are given a nonlinear optimization primal problem (P). Suppose that the dual problem (D) to the primal optimization problem (P) achieves a zero duality gap with a solution to the primal ...