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3
votes
1answer
36 views

Trouble seeing why this is the dual of an LP

$A$ is an $m \times n$ matrix. Using the notation $x=(x_1, \ldots, x_n)$, $z=(z_1, \ldots, z_m)$, and $y=(y_1, \ldots, y_m)$, I'm reading that if the primal LP is $$ \min 0x_1 + 0x_2 + \cdots + 0x_n ...
1
vote
1answer
36 views

Functional defined on the space of functions with compact support

Let $X=C_c(\mathbb{R})$, the space of functions with compact support, normed with the max norm. Define $\Gamma: X \rightarrow \mathbb{R}$ as: $$ \Gamma f= \int_{- \infty}^{\infty} f(t)dt ...
0
votes
0answers
32 views

Dual of a polyhedra vs. dual of an optimalization problem

There are lot of fields where the term duality appear. Is there any relationship between dual of an optimalization problem and dual of a polyhedra?
2
votes
1answer
49 views

Hecke equivariance in Poincaré duality.

Consider the first singular homology and cohomology groups of a modular curve, $H^1(X,\mathbb{Z})$ and $H_1(X,\mathbb{Z})$. The Hecke algebra acts on both of them and they are dual to each other under ...
0
votes
0answers
17 views

Dual representations of affine span of a set of affine transformers.

I am not well versed in the literature of affine transformers or Farkas' Lemma. I just know the basics of the two concepts. Are there any dual representation of affine span of a set of affine ...
5
votes
1answer
82 views

Categorical proof of Pontrjagin Duality?

I would like to ask if there is any reference in which Pontrjagin Duality is proved in a categorical context: I started reading the Pontrjagin Dual entry in nLab, ...
0
votes
0answers
29 views

dual feasibility question in augmented Lagrangians and the method of multipliers

I am going through Boyd's tutorial on ADMM. My question is basically from Sec 2.3. Consider the optimization problem $$\min.~f(x)~~~~\text{s.t.}~~~Ax = b.$$ Then the Lagrangian is ...
1
vote
0answers
47 views

Dual Vector Spaces - Evaluation at a Point of a Polynomial Gives a Basis

My question is the following: Let $\{a_0,a_1,...,a_n\}$ be (pairwise) distinct, real numbers. Let $V$ be the vector spaces of all polynomials of degree at most $n$, ie $V = \Bbb P_n$. Let $\phi_j : ...
3
votes
0answers
43 views

Understanding the duals of $L^{\infty}(K)$ and $C(K)$ and weak-* compactness

Let $(K\subset \mathbb{R}^n,\mathcal{B}(K),\lambda)$ be a measure space, where $\lambda$ is the Lebesgue measure and $K$ is compact. According to Wikipedia (with adapted notation), The dual ...
1
vote
1answer
722 views

Dual Simplex Method Example Problem

I have tried to solve this Linear Program: max z = −2*x1 − x2 s.t. −2*x1 + x2 + x3 ≤ −4 x1 + 2x2 − x3 ≤ −6 x1,x2,x3≥0 Choosing -6, ...
1
vote
1answer
66 views

The first Weyl algebra is Calabi-Yau

Why is the Weyl algebra $A_1(k)$ over a field $k$ Calabi-Yau? (My definition of Calabi-Yau is Ginzburg's)
2
votes
2answers
79 views

Dual Vector Space embedding

Is there an embedding of any vector space $V$ into $V^*$? As far as I know it is not true. The statement that I know of is that there is natural embedding of $V$ into $V^{**}$ Is there any ...
2
votes
2answers
92 views

Dual Mapping Preserves Linear Independence if and only if Original Mapping is Surjective

Here is my question: Let $V$ and $W$ be finite-dimensional vectors spaces over a field $F$ and $f:V \rightarrow W$ a linear map. Show that $f$ is surjective if and only if the image under ...
2
votes
0answers
32 views

Reference Request: Algebraic Serre's Duality Theorem for Curves

Serre's Duality Theorem is well known and well studied and, as far as I know, there is a "big" algebraic proof for the general case, which is now kind of standard, and can be found in Hartshorne and ...
1
vote
0answers
26 views

Prove one program to be another's dual program

This is the Primal problem $\min f_0(x)$, subject to $f_i(x) \leq 0$, i = $1,2,\ldots,m$ $f(x)$ is a linear program and my target is to prove that $\max g(λ)$ subject to ...
1
vote
0answers
27 views

dual basis and standaard basis and matrix

let $V$ be a vector space from dimantion $n$ and $V^{\star}$ be a map from $V$ to $R$ ($V^{\star}$: $V$$\mapsto$$R$) and $A$ be a matrix from a bilineare form $T$:$V$$\times$V$^{\star}$ $\mapsto$$R$ ...
3
votes
1answer
118 views

Show that $L^1\subsetneq (L^\infty)^*$ [duplicate]

How does one show that $L^1\subsetneq (L^\infty)^*$? I am having trouble in this. Any help would be appreciated.
0
votes
0answers
29 views

flabbiness of hyperfunctions

Let $A(\mathbb R^n)$ be the real analytic functions and $\mathscr B(\mathbb R^n)$ the hyperfunctions, dual to $A(\mathbb R^n)$. Further let $W\subset \mathbb C$ be a cone in the complex plane with ...
0
votes
1answer
89 views

Natural map from vector space to double dual not surjective example

I recognize the fact that the natural map from an infinite dimension vector space $V$ to it's double dual space $V^{**}$ need not necessarily to be surjective because we don't have that the $\dim V$ = ...
0
votes
1answer
94 views

Maximization with the Dual using the Simplex Method.

I have an exam in a few hours. I need to understand the solution to the following question Find the Maximal to the the following $2 x_1 + 3 x_2$ is the objective function. The constraints are ...
1
vote
1answer
45 views

How to obtain primal problem from Lagrangian?

If you're trying to optimize $\min_x f_0(x)$ subject to $f_i(x) \leq 0$ then the Lagrangian would be $$L(x, \lambda) = f_0(x) + \sum_i \lambda_i f_i(x)$$ The dual problem is $\max_\lambda g(y)$ ...
1
vote
1answer
80 views

Existence of a function in Radon Measure

I am having trouble to construct a function $f$ in the following problem. Let $X$ be Locally Compact Hausdorff topological space. Let $\mu$ is a positive Radon measure on $X$ with $\mu(X)=\infty$ ...
3
votes
1answer
80 views

Proof of Poincare duality in Bott and Tu's book

In Bott and Tu's book, "Differential forms in Algebraic Topology", page 45, Section 5 of Chapter one, he tried to prove the Poincare duality. But I find one step mysterious, namely when he describes ...
3
votes
1answer
25 views

cofree comodules and embedding

For an $R$-coalgebra C, is it possible for every C-comodule M to be embeded into a C-comodule of the form $\underset{i \in I}{\bigoplus} C$?
-1
votes
1answer
92 views

Proof of properties of dual cone

Show that if C $\subseteq$ D then $D^*$ $\subseteq$ $C^*$ where * is dual cone operation. Can somebody explain it.
1
vote
1answer
27 views

Calculating the dual module

Is the dual $\mathbb{Z}$-module of $\mathbb{Z}/n\mathbb{Z}$, that is ${\rm Hom}_{\mathbb{Z}}(\mathbb{Z}/n\mathbb{Z},\mathbb{Z})$ isomorphic to $\mathbb{Z}/n\mathbb{Z}$. Looking at it briefly I think ...
0
votes
1answer
102 views

The dual space of normed vector space $X$ is isomorphic to the dual of its completion

Let $\overline{X}$ be Banach space and $X$ be dense subset of it. Show that dual space of X and dual space of $\overline{X}$ are isomorphic. Why these are isomorphic? I don't know how to prove ...
4
votes
1answer
104 views

Duality in algebraic de Rham cohomology

I am trying to prove that the following is a short exact sequence $$ 0 \rightarrow H^0(X,\Omega_X) \rightarrow H^1_{\text {dR}}(X/k) \rightarrow H^1(X,\mathcal O_X) \rightarrow 0, $$ where $k$ is an ...
5
votes
2answers
73 views

Constructing Dual notions via Duality

First of all, I do not have much mathematical background and I have minimal category theory knowledge. I am just trying to understand one or two things about category theory because the concept sounds ...
2
votes
1answer
73 views

Auto-Langlands dual gruops.

Consider a semisimple Lie group $G$. We define the Langlands dual $\hat{G}$ of $G$ as the group which has as a root system, the root system generated by the coroots of $G$. Recall that given a root ...
0
votes
1answer
24 views

Duality in a finite dimension

Let $E$ and $F$ be two finite dimension vector spaces over the same field $K$, $V$ is sub-space of $E$, $L_V(E,F)$ is the set of linear maps from $E$ to $F$ which vanish on $V$. And let $W$ such that ...
2
votes
1answer
116 views

The convex conjugate of a quadratic form with positive semi definite matrix

I want to find the convex conjugate (Legendre transform) of a quadratic for $1/2x^{t}Qx$ when $Q$ is positive semi-definite. If Q is non-singular, the solution is easy - the gradient is $y-Qx=0$ so ...
2
votes
0answers
35 views

Is this dual-spaced norm based on $L_2$ norm

I am reading the book of Claes Johnson about Numerical Solution of Partial Differential Equations by the Finite Element Method and particularly pages 34 and 98. I wrote these notes to my craft Is ...
3
votes
1answer
137 views

Lagrange dual of a sum of convex functions

Given a set of convex functions $f_i(x)$ and convex sets $X_i$ in $\mathbb R^n$ I need to find the Lagrange dual problem for the problem $\min \sum{f_i(x)} , x \in X_i \forall i$. There is of course ...
0
votes
1answer
81 views

Lagrangian dual for the sum of norms

I would like some help in deriving the Lagrangian dual function of a sum-of-norms minimization problem : $\sum{||A_{i}x-b_{i}||}$ when $A_{i}$ are matrices, and $b_{i},x$ vectors. I understand I can ...
1
vote
1answer
387 views

Orthogonal Complements property

I have a question about how to prove a certain property of orthogonal complements of vector subspaces. Given $\mathrm E$, a vector space over a commutative field k, define: $$\phi\text{ : } \mathrm E ...
3
votes
2answers
282 views

Recovering the solution of optimization problem from the dual problem

In the context of (most of the times convex) optimization problems - I understand that I can build a Lagrange dual problem and assuming I know there is strong duality (no gap) I can find the optimum ...
0
votes
0answers
25 views

how to get the dual of that optimization problem

max $1^\top x$ such that $x^\top M x = 0$ and $x_i^2 = x_i$ for all i For the above problem how can I derive the dual form. My main problem is to choose matrix notation or the element-wise notation ...
0
votes
1answer
111 views

how to construct the Lagrangian dual problem?

The primal optimization problem is, \begin{align*}\min_x\;&f_0(x)\\ \text{s.t.}\;&f_i(x)\le0\\ &h_j(x)=0\end{align*}, to construct the dual problem, I form the Lagrangian, ...
0
votes
1answer
76 views

Dual space of a complete vector space

Let $(X, \mathcal{F}, \mathbb{P})$ be a Probability space. Consider the space of all functions with topology induced from convergence in Probability. I am interested in knowing the dual space of it. ...
2
votes
1answer
47 views

Riesz Representation Theorem: isomorph

Riesz' Representation Theorem states that every linear functional can be represented by a vector. This shows that the Dual can be ANTILINEARLY and norm preserving identified with the Hilbert Space ...
1
vote
2answers
74 views

How to find dual spaces?

I would greatly appreciate it if you could kindly share how to find dual spaces? For example, let X be the vector space of n-dimensional vectors with the Euclidean norm. Prove that X*=X. I know a ...
0
votes
1answer
93 views

Linear isoparametrics with dual finite elements

The subject presented here is some content of the Wikipedia page about Platonic solids combined with my own experience on Finite Elements.To start with the latter, there is a standard piece of Finite ...
1
vote
1answer
99 views

What is dual representation in plain English?

Can someone please explain what is Dual representation in plain English. I read its definition on wikipedia and at many other places but could not develop an intution for it. Please explain in plain ...
2
votes
1answer
206 views

Serre duality as a right adjoint functor

As stated on the wikipedia page, Grothendieck generalized Serre duality by stating that there exists a right adjoint functor $f^!$ to the functor $Rf_!$ when one works within the correct category. ...
1
vote
1answer
82 views

Dual of this primal optimization problem?

How would one find the dual of the following problem? $min_x 1/2 ||y-x||_2^2 + \lambda||x||_1 $ Can someone please explain to me how to do this since there are no specific constraints?
5
votes
1answer
427 views

Dual to the dual norm is the original norm (?)

I have the following questions about dual norms : How do you prove that the dual of the dual norm is in fact the original norm? This is what I have so far: If I have $\|y\|_* $ as the norm dual of ...
1
vote
1answer
58 views

inf sup duality in Hilbert spaces

Let $Y$ be a Hilbert space, for all $y \in Y$ and $X$ a closed subspace of $Y$, I want to prove the following duality result: $$\inf_{g \in X} || y -g|| = \sup_{(f,X)=0} \frac{(y,f)}{||f||},$$ where ...
1
vote
1answer
1k views

Is it logically valid to prove DeMorgan's laws using the duality of boolean algebra?

I'm taking an introductory course in boolean algebra, and have been assigned the task of proving DeMorgan's Laws (so, disclaimer, this is homework). One line of reasoning that I came up with would be ...
2
votes
2answers
143 views

Properties of the duals of $\ell^1$ and $\ell^{\infty}$

a) True or false: (i) $(\ell^{1})^* = \ell^{\infty}$ (ii) $\ell^1 \subset (\ell^\infty)^*$ (iii) $(\ell^\infty)^* \subset \ell^1$ (iv) $(\ell^1)^{**} \subset \ell^1$ b) Give the set of dual vectors ...