Questions tagged [duality-theorems]

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

In mathematics, the term "dual" may have several meaning:

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Motivation to understand double dual space

I am helping my brother with Linear Algebra. I am not able to motivate him to understand what double dual space is. Is there a nice way of explaining the concept? Thanks for your advices, examples and theories.
Theorem
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Precise connection between Poincare Duality and Serre Duality

The statements of Poincare duality for manifolds and Serre Duality for coherent sheaves on algebraic varieties or analytic spaces look tantalizingly similar. I have heard tangential statements from some people that there is indeed some connection…
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What is duality?

I have seen some examples of duality. Sometimes applied to theorems, as for example Desargues theorem and Pappus theorem. Sometimes applied to spaces, for example the dual space of a vector space. Sometimes even applied to a method like simplex and…
Ambesh
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What can be said about the dual space of an infinite-dimensional real vector space?

What can be said about the dual space of an infinite-dimensional real vector space? Are both spaces isomorphic? Or shall we have something like the dual of the dual is isomorphic to the initial vector space (same as with the perpendicular subspace…
El Moro
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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 "theorem is "dual"". Often people say "dual comes for free". Like you swap variables or something and you get another…
bodacydo
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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 $$J: Y \to Y'', J(y)=j_{y}, j_{y}(y')=y'(y)$$, where $Y''$ denotes the bidual space of…
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Motivation for Mirror-Descent

I am trying to wrap my head around why mirror descent is such a popular optimization algorithm. Based on my reading, it seems like the main reason is that it improves upon the convergence rate of subgradient descent, while only using full gradient…
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Fenchel dual vs Lagrange dual

Consider the Fenchel dual and the Lagrangian dual. Are these duals equivalent? In other words, is using one of the these duals (say for solving an optimization), would give the same answer as using the other one? I think the answer is no, but I am…
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Pontryagin duality for finite groups

Let $\mathbb{Q}$ denote the group of rational numbers (with addition as the binary operation) and let $\mathbb{Z}$ denote the subgroup of integers. The Pontryagin dual of a group $G$ is the group $G^* = \operatorname{Hom}(G,\mathbb{Q/Z})$. (It is…
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What are some theorems made easier by Stone Duality?

I have seen a lot of praise for the Stone Duality Theorem, which links the algebraic structure of boolean algebras to the topological structure of stone spaces by a (contravariant) adjoint equivalence of categories. What are some theorems which are…
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What is the relation between dual spaces and inner product?

What is the relation (if any) between dual spaces and inner product? As far as I understand the dual space of a vector space is the set of all linear mappings from the vector set to the field over which the space is defined. But the definition of…
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Hilbert-valued Schwartz functions

Let $H$ be a separable complex Hilbert space. We can define Schwartz functions $f\colon\mathbb R^n\to H$ to be the smooth functions for which $$ \sup_{x\in\mathbb R^n}\|(1+|x|^2)^mD^\alpha f(x)\|_H<\infty $$ for all $m\in\mathbb N$ and all…
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Dual and completion of metric spaces

Say we have a metric space $(M,d)$, and we want to complete it in the following sense: Definition: A completion of $(M,d)$ is a complete metric space $(\widetilde{M},d')$ together with a Lipschitz funcion $i:M\rightarrow\widetilde{M}$ such that for…
Luiz Cordeiro
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Is KKT conditions necessary and sufficient for any convex problems?

In Boyd's Convex Optimization, pp. 243, for any optimization problem ... for which strong duality obtains, any pair of primal and dual optimal points must satisfy the KKT conditions i.e. $\mathrm{strong ~ duality} \implies \mathrm{KKT ~ is ~…
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Help me organize these concepts -- KKT conditions and dual problem

This is a long question in which I explain my current understanding of certain ideas. If anyone is interested in reading this and would like to provide any commentary/feedback that may help me understand these ideas more clearly, or that you think…
littleO
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