# Questions tagged [duality]

Use for questions regarding duality of mathematical object, i.e. dual spaces, objects with two possible interpretations etc.

224
questions

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102 views

### Reference for duality inducing bijections between subobjects and quotients?

I'm not the most category-theoretic person but I have run into the following statement in my work. Suppose we have a category duality $\mathcal F:\mathcal C\to\mathcal D$. For my needs you can ...

**9**

votes

**0**answers

100 views

### Finitely presentable grids and co-presentable topological spaces

In the 90s there has been some interest for the category $\mathsf{Top}^\circ,$ the dual of the category of spaces. Most of the relevant papers on the topic are co-authored by Pedicchio.
Barr, ...

**6**

votes

**1**answer

157 views

### Top cohomology of profinite Poincaré duality group

The paper "Cohomology of p-adic analytic groups" by Symonds and Weigel is considered one of the main references for continuous cohomology of profinite groups. There is a passage I do not ...

**35**

votes

**12**answers

3k views

### No canonical isomorphism [duplicate]

I thought that it would be interesting to collect into a big list various instances of isomorphic structures with no preferred isomorphism between them. I expect the examples to be interesting since ...

**5**

votes

**1**answer

210 views

### Generalization of Bernstein’s inequality

I'm using Muscalu and Schlag's textbook to study harmonic analysis and I encountered the following claim:
Given some function $f \in \mathcal{S}(\mathbb{R}^{d})$, where $\mathcal{S}(\mathbb{R}^{d})$ ...

**1**

vote

**1**answer

66 views

### Confusing definition of homogeneous Sobolev norm of order -1

Let $\Omega \subset \mathbb{R}^{d}$ and $\|.\|$ is the standard euclidean $2$-norm. I came across a definition of $\dot{H}^{-1}(\Omega)$ which is a bit confusing. In [1] authors define the following ...

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votes

**0**answers

172 views

### Maxwell $U(1)$ gauge theory's electric and magnetic sources turned on simultaneously in the classical differential geometry

Question:
How do we couple $U(1)$ electric (E) and magnetic (M) sources simultaneously in the classical differential geometry language, in a $U(1)$ gauge theory based on $U(1)$ gauge bundle and its $U(...

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35 views

### Dual of union of Reproducing Kernel Hilbert Spaces

I have a union of Reproducing Kernel Hilbert Spaces $\mathcal{B}$. I am interested in finding the dual of $\mathcal{B}$. Knowing what the dual is might help to write an alternate formulation for the ...

**4**

votes

**0**answers

91 views

### Wasserstein distance and Monge-Kantorovich-Rubinstein duality

The definition of Wasserstein $p$-distance between two measures $\mu$ and $\nu$ on a Polish space $X$ is given by
$$
W_p^p(\mu, \nu) = \inf_{\gamma \in \Pi(\mu, \nu)} \int_{X \times X} d(x, y)^p\; d\...

**10**

votes

**1**answer

255 views

### Origin and context of adjunctions inducing equivalences between full subcategories

The following is well-known.
Theorem. Let $F\dashv U$ be a pair of adjoint functors
$$F\colon \mathcal C\to \mathcal D, \qquad U\colon \mathcal D\to\mathcal C$$
with unit $(\eta_A\colon A\to U(F(A)))_{...

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votes

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46 views

### Conjugate of composition in Bochner spaces

Let $H$ be a separable Hilbert space (of non-zero dimension), let $(\Omega,\Sigma,\mu)$ be a finite measure space, and let $L^2(\mu;H)$ be the Bochner-space $\mu$-integrable $H$-valued functions. ...

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40 views

### Dual cone operation is an antitone morphism of lattices

Currently I'm reading a classical text Convex Sets by F. Valentine. While reasing it I'm trying to generalize results as much as I can. The chapter on Dual cones made me thinkin of the fallowing ...

**8**

votes

**1**answer

175 views

### Comparison: Formal Wirthmüller isomorphism of Fausk-Hu-May vs. Balmer et. al

$\newcommand{\Cc}{\mathcal{C}}$
$\newcommand{\Dd}{\mathcal{D}}$
$\newcommand{\tensor}{\otimes}$
$\DeclareMathOperator{\Sp}{Sp}$
This question is about comparing the approaches for a formal Wirthmüller ...

**4**

votes

**1**answer

144 views

### Can one characterize maximal antichains in terms of distributive lattices?

This is inspired by the recent question Verification of a maximal antichain
The celebrated duality between finite posets and finite distributive lattices has several nice formulations. One of them ...

**13**

votes

**1**answer

480 views

### Are Spanier-Whitehead duals of general spaces expressible through some generalization of normal bundles?

The question is inspired by an answer to The concept of Duality
It is explained in that answer that the Spanier-Whitehead dual of a compact manifold is given by the Thom spectrum of normal bundles of ...

**3**

votes

**1**answer

139 views

### On the dimension of the dual variety of a singular hypersurface

I was primarily interested in the following question. Let $n\geq 3$, and let $X\subset \mathbb{P}^n$ be a degree $d$ hypersurface. Assume that its singularity locus $S$ (with reduced structure) is ...

**3**

votes

**1**answer

118 views

### Duality argument for elliptic regularity

M. Dauge proved in [1] the regularity property "$\Delta u \in (W^1_{p'})^*$ $\Rightarrow$ $u \in W^1_p$" for Dirichlet and Neumann problem in domains with piecewise smooth boundaries, for $p&...

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votes

**1**answer

146 views

### Is every sequentially $\sigma(E',E)$-continuous linear functional on a dual Banach space $E'$ necessarily a point evaluation?

$\newcommand{\bf}[1]{\mathbb #1}\newcommand{\sc}[1]{\mathscr #1}$
A duality between two vector spaces $E$ and $F$ over $\bf K$ ($= {\bf R}$ of ${\bf C}$)
is, by definition, a bilinear form
$$
\...

**8**

votes

**1**answer

185 views

### Are there centrally-symmetric self-dual polytopes in dimension $d> 4$?

A convex polytope $P\subset\Bbb R^d$ is centrally symmetric if $-P=P$. It is self-dual (or better, self-polar?) if its polar dual $P^\circ$ is congruent to $P$, that is, there is a map $X\in\mathrm O(\...

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vote

**1**answer

54 views

### Calculating vertex weights

Vertex weights are a metaphor for a constant value $\pi_i$ that is added to the weight of every edge $e_{ij}$ that is adjacent to vertex $v_i$ in a symmetric graph $G(V,E)$ with weighted edges.
The ...

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vote

**1**answer

65 views

### Weak duality sign

Let $\left\{\begin{matrix}
\operatorname{min}_xc^Tx\\Ax\leq b
\\ x\in \mathbb{R}^+
\end{matrix}\right.$ be a LP primal problem and $\left\{\begin{matrix}
\operatorname{max}_yb^Ty\\A^Ty\geq c
\\ y\in ...

**1**

vote

**3**answers

262 views

### Duality of finite signed measures and bounded continuous functions

Let $E$ be a metric space, $C_b(E)$ denote the space of bounded continuous functions $E\to\mathbb R$ (equipped with the supremum norm), $\mathcal M(E)$ denote the space of finite signed measures on ...

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**0**answers

55 views

### From $f$-divergence to its dual: the transformation of convex functions on $\mathbb R_+$ by $f^*(t) = 1 f(\frac 1 t)$

I would like to understand the relationship between minimising the KL divergence $P \mapsto D_{KL}[P,Q]$ and the reverse KL divergence $P\mapsto D_{KL}^*[P,Q]=D_{KL}[Q,P]$ for probability measures $P$ ...

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34 views

### Self-duality of cones associated with elementary symmetric polynomials

Let $n\ge3$ be an integer, and denote $\sigma_1,\ldots,\sigma_n$ the elementary symmetric polynomials in $n$ indeterminates:
$$\sigma_1(X)=X_1+\cdots+X_n,\quad\ldots\quad,\sigma_n(X)=X_1\cdots X_n.$$
...

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68 views

### Preservation of the Euler-Poincaré characteristics and commutativity of the multiplication in R^d

While talking about duality for polyhedra and the analogue for polygons (where the duality is essentially trivial) with a friend, I wondered if this self-duality in dimension 2 was anyhow related to ...

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62 views

### A strong duality for convex functional optimization that admits Lipschitz continuity constraints?

Problem Statement
I am looking for formal proof---hopefully textbook material---of two items:
an analogue to Slater's condition [1] that obtains strong duality for optimization of convex functionals; ...

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vote

**1**answer

125 views

### Dual problem with integrals

I am reading a paper where the author derives the following Lagrangian dual problem :
$\min_v \int_R \frac{1}{4} \frac{\beta^2}{v-2\|x\|}dx+v\;\;\;\text{s.t.}\;\;\;v\geq 2\|x\|\;\;\;\forall x \in R$
...

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20 views

### Weak relaxation of a strongly lower semi-continuous functional

Let $F$ be a lower semicontinuous functional on a Banach space $X$, wrt its strong topology. Is there a known form for the relaxation (lower semicontinuous envelope) of $F$ with respect to the weak ...

**3**

votes

**2**answers

270 views

### Bi-annihilator of a subspace of the dual of an infinite-dimensional vector space

Let $V$ be an infinite-dimensional vector space and $V^*$ its dual.
For a linear subspace $W\subset V$ define $W^ \circ\subset V^*$ as the subspace of linear forms on $V$ vanishing on $W$.
Dually, for ...

**5**

votes

**0**answers

190 views

### Riesz Representation Theorem for $L^2(\mathbb{R}) \oplus L^2(\mathbb{T})$?

The spaces $L^2(\mathbb{R})$ (square-integrable functions) and $L^2(\mathbb{T})$ (1-periodic square-integrable functions, considered over the real line $\mathbb{R}$) are two subspaces of the space of ...

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87 views

### When is the natural map of Tate cohomology an isomorphism?

First of all I want to say that I am not at all an expert in Group cohomology . Recently I attended a seminar where the speaker mentioned about something called Tate cohomology groups which in ...

**3**

votes

**1**answer

96 views

### Duality problem of an infinite dimensional optimization problem

I am reading the paper "OPTIMAL INEQUALITIES IN PROBABILITY THEORY: A CONVEX OPTIMIZATION APPROACH" by BERTSIMAS and POPESCU. In the paper, the authors derived a duality problem for an ...

**0**

votes

**1**answer

68 views

### If $\tau_1\subset \tau_2$ and $X^*$ is separable for $\tau_1$ then $X^*$ is separable for $\tau_2$?

Let $X$ be a Banach space the associated dual space is denoted by $X^*$. Take $\tau_1$ and $\tau_2$ two topologies in $X^*$ compatible with the duality $(X^*,X)$, such that $\tau_1\subset \tau_2$.
...

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103 views

### Dual space of a periodic Sobolev space $H^m([0,2\pi]^n)$

For $m \in \mathbb{N}$ let $H^m([0,2\pi]^n)$ be the periodic Sobolev space on $\mathbb T^n=\mathbb R^n/(2\pi\mathbb Z^n)$ equipped with the norm
$$
\|u\|_{H^m} = \left(\sum_{k \in \mathbb{Z}^n} (1+|k|^...

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82 views

### Dual space of mean-free Sobolev space

I am considering the space $V:=\{v \in H^1(\Omega): \int_\Omega v = 0\}$ of mean free functions. What is the dual space of this space? Is the dual space given by $D:= \{f \in (H^1(\Omega))^*: \langle ...

**5**

votes

**1**answer

283 views

### A generalization of integral Poincaré duality

In this paper, Felix, Halperin and Thomas define the notion of a Gorenstein space over a field $\mathbb{k}$:
An augmented differential graded algebra $R$ over $\mathbb{k}$ is Gorenstein if $\text{Ext}...

**6**

votes

**1**answer

184 views

### If $X$ is separable space then $X^∗$ is separable in all topologies $\tau$ such that $(X^∗,\tau)^∗ =X$?

Let $(X,\|.\|_{X})$ be a separable Banach space and the associated dual space is denoted
by $X^*$. By $w^*$ we shall indicate the weak$-*$ topology on $X^*$.
Let $B_{X^∗}= \{x^∗ \in X^∗ : \|x^∗\|_{X^∗...

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154 views

### Dual varieties and nodal sections

Let $X$ be a(n even dimensional) smooth complex projective variety in $\mathbb{P}^N$, and let $X^{\vee}$ be its dual variety; up to an higher degree Veronese embedding of $X$, I assume that $X^{\vee}$ ...

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127 views

### Do Poincaré duality algebras need to be defined over a field?

I asked the below question here on MSE, but after some time and a bounty offering I have not received an answer.
A graded commutative, connected $\mathbb{k}$-algebra $A$ is called a Poincaré duality ...

**1**

vote

**1**answer

248 views

### Poincare duality-differential geometry

Let $ M $ be a smooth and compact manifold with boundary $\partial M = X \times F $ on which the structure of a smooth locally trivial bundle $$ \pi: \partial M \longrightarrow X $$
where the $ X $ ...

**15**

votes

**1**answer

779 views

### What is a module over a Boolean ring?

Recall that a (unital) Boolean ring is a (unital) commutative ring $A$ where every element is idempotent; it follows that $A$ is of characteristic 2. There is an equivalence of categories between ...

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263 views

### Does any 'logical' theory have a bounded ∞-pretopos as syntactic category?

Stone duality may be understood as providing a duality between syntax and semantics for propositional logic, so that a theory may be recovered from its models. In order to do likewise for first-order ...

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**0**answers

51 views

### Proving the existence of a dual for an infinite linear program

I am concerned with proving the existence of the dual of an infinite linear program. In addition to the writings of Rockafellar, Luenberger, and Boyd & Vandenberghe on: subdifferentials, Legendre-...

**4**

votes

**1**answer

331 views

### Which topological spaces admit embeddings into Euclidean spaces

I'm interested in the dual question to:
continuous images of open intervals, about surjections onto open intervals.
Namely, if $X$ is a topological space, when can we guarantee that there exists a ...

**2**

votes

**1**answer

320 views

### Why are Serre functors always exact?

Let $k$ be a field and $\mathcal{T}$ be a $k$-linear triangulated category with finite dimensional spaces of morphisms. Bondal and Kapranov proved that every Serre functor on $\mathcal{T}$ is exact (...

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votes

**2**answers

206 views

### Completeness of coefficient functionnals

My questions is about Schauder bases and more specifically about coefficient functionals.
Let $(x_n)$ be a Schauder basis of a Banach space $X$. Thus for all $x$ in $X$, $x = \sum f_n(x) x_n$. The $...

**8**

votes

**1**answer

317 views

### Dual space of continuous Banach-space-valued functions

Let $X$ be a Banach space and $K$ some compact Hausdorff space. I am interested in the dual space of the Banach space
$$C(K; X) = \lbrace f: K \to X, \ f \text{ is continuous}\rbrace, \qquad \lVert ...

**1**

vote

**2**answers

184 views

### Is unit ball in 2-Wassestein metric weakly compact?

This might be a trivial question, but I am trying to prove equipment-coerciveness of some family of functions on the space of Probability measures on some space. I could reduce the problem to showing ...

**2**

votes

**1**answer

120 views

### Characteristic polynomial of the line graph (originally dual graph)

I am quite sure I have seen somewhere the connection between the characteristic polynomial of a (finite undirected) graph and its dual. I am not able to find it currently. Could you please refer me to ...

**1**

vote

**0**answers

215 views

### Continuity of the Legendre transform of a Lipschitz function

Let $p > 1$, $f$ defined on $L^p(\mathbb{R}^n,\mathbb{R})$, locally Lipschitz and concave, with $f(x) = 0$ when $x \geq 0$ a.e. We define, with $q$ dual to $p$, for any $y\in L^q$ , $g(y) := \sup_{...