Questions tagged [duality]

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

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Weak*-separability of the unit ball of $X’$ and density characters and cardinalities of $X$ and $X’$

(This question has also been asked on Math StackExchange.) Let $X$ be a Banach space, $X’$ be its continuous dual such that its unit ball is weak*-separable. I’ve been wondering what can be said about ...
David Petey Gao's user avatar
5 votes
1 answer
116 views

Complemented subspaces of a dual Banach space

Let $\kappa$ be an infinite cardinal number and by $\mathcal{B}(\kappa)$ denote the class of all Banach spaces of density $\kappa$. My question reads as follows: Does there exist $\kappa$ for which ...
Damian Sobota's user avatar
1 vote
1 answer
133 views

Homology of complements and homotopy equivalence

Albrecht Dold gave a short proof of the Jordan-Alexander complement theorem in the following form. Given two closed sets $A$, $B \subset {\bf R}^n$ that are homeomorphic, the singular homology groups $...
coudy's user avatar
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1 vote
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$BMO$ is not reflexive

It is well known that $BMO$ is the dual space of the Hardy space $H^1$, which is the dual space of $VMO$. I believe that $BMO$ is not reflexive, but I am not quite sure that the above information is ...
Bazin's user avatar
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1 vote
3 answers
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Given a compact set $K \subset \mathbb{R}^n$, is the space of distributions supported on $K$ the dual of some test function space?

I am aware that the dual of $C^\infty(\mathbb{R}^n)$ is the space of distributions (not necessarily tempered) with compact support. However, if we fix a compact set $K \subset \mathbb{R}^n$, is the ...
Isaac's user avatar
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4 votes
0 answers
376 views

Fixing the duality $L^\infty(X)= L^1(X)^*$ for Radon measure spaces

Consider the following fragment from Folland's book "A course in abstract harmonic analysis": Let me denote the Borel subsets of $X$ by $\mathscr{B}(X)$. Folland claims that if $\mu$ is a ...
Andromeda's user avatar
1 vote
1 answer
62 views

Legendre transformation of vector valued function

Good afternoon. Is there any generalisation of Legendre--Fenchel transformation to the vector-valued functions $f: \mathbb{R}^n \to \mathbb{R}^n$?
Dmitry Vilensky's user avatar
6 votes
0 answers
131 views

Explicit computations of Serre duality for elliptic curves

I have an elliptic curve $E$ defined over a ring $R$, I want to compute the pairing $$ H^1(E,\mathcal{O}_E)\times H^0(E, \Omega_E^1){\rightarrow}R. $$ Clearly we have that $H^0(E, \Omega_E^1)=R \...
marco's user avatar
  • 79
4 votes
0 answers
61 views

Duality for finite quotient groups of finitely generated free abelian groups

$\newcommand{\Z}{{\Bbb Z}} \newcommand{\Q}{{\Bbb Q}} \newcommand{\Hom}{{\rm Hom}} $ The following lemma is certainly known. Lemma (well-known). Let $B$ be a lattice (that is, a finitely generated ...
Mikhail Borovoi's user avatar
2 votes
0 answers
68 views

Is it possible to deduce Poincaré duality from duality of polytopes?

I'm having trouble understanding Poincaré duality, as it seems unmotivated. Here for instance: https://math.stackexchange.com/a/14469/454016 Poincaré duality is explained through a duality of ...
Alexander Praehauser's user avatar
8 votes
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247 views

Isbell duality between algebras and sheaves

nLab says on Isbell duality, the following: A general abstract adjunction $(\mathcal{O} \dashv \operatorname{Spec}) : \mathrm{CoPresheaves} \leftrightarrows \mathrm{Presheaves}$ relates (higher) ...
Timo's user avatar
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3 votes
1 answer
123 views

Riesz representation theorem for duals of spaces of continuously differentiable functions

Let $k$ be a positive integer. I am looking for a possibly exhaustive reference discussing representation of dual spaces of $C^k_b(\mathbb{R}^d)$, $C^k_0(\mathbb{R}^d)$, or at least $C^k(K)$ for ...
ajr's user avatar
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1 vote
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Is the Lie bracket on $\mathfrak g^{\ast}$ induced from a cocommutator defined on $\mathfrak g\ $?

Let $G$ be a Poisson-Lie group. Let $\mathfrak g = \text {Lie} (G) = T_1 G$ be the corresponding Lie algebra. Then the Poisson structure on $G$ gives rise to a Lie bracket $[\cdot, \cdot]$ on $\...
Anil Bagchi.'s user avatar
5 votes
0 answers
114 views

Faltings' Cartier duality for A-modules in terms of Hopf algebras

$\newcommand\dual{^{\text{dual}}}\newcommand\GrpSch{\mathrm{GrpSch}}\DeclareMathOperator\Hom{Hom}\DeclareMathOperator\Vect{Vect}$If $G$ is a finite group scheme over a field $k$, we can define its ...
Homotopy theorist 's user avatar
8 votes
1 answer
209 views

Finite domination and Poincaré duality spaces

Here are some definitions: A space is homotopy finite if it is homotopy equivalent to a finite CW complex. A space finitely dominated if it is a retract of a homotopy finite space. A space $X$ is a ...
John Klein's user avatar
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$\ast$-autonomous categories with non-invertible dualizing object?

1. Definition Firstly, recall the following nLab-definition of a $\ast$-autonomous category: A $\ast$-autonomous category is a symmetric closed monoidal category $(C,\otimes,I,\multimap)$ with a ...
M.C.'s user avatar
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1 vote
0 answers
35 views

Application of the $\operatorname{BMO}$, $H^1$ duality

Let $f\in \operatorname{BMO}(\partial \Delta)$, then there exists a Carleson measure $\mu$ in $\Delta$ such that $$f(\zeta)-\int_{\Delta}P_{z}(\zeta)d\mu(z)\in L^{\infty}(\partial \Delta),\ \zeta\in\...
Ferry Tau's user avatar
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3 votes
1 answer
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When Alexander dual of a simplicial complex is a matroid?

Let $C$ be a simplicial complex on a finite set $V$: that means $C$ is a collection of subsets of $V$ such that if $\sigma\in C$ and $\tau\subseteq \sigma$, then $\tau\in C$. The Alexander dual $D(C)$ ...
Connor's user avatar
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173 views

Spanier-Whitehead dual of space of natural transformations

Let $F, G: \mathcal{J} \to \mathsf{Sp}$ be continuous functors between $\sf{Sp}$-enriched categories, where $\sf{Sp}$ denotes any of the point-set models for spectra (i.e., orthogonal spectra). ...
stableunknown's user avatar
0 votes
0 answers
62 views

solution of equivalent problem Kantorovich for case squared distance function

We know that the Kantorovich duality when the cost function is the square Euclidean distance is equivalent to $$ \inf_{(\tilde\varphi,\tilde\psi)\in \tilde\Phi_c} J(\tilde\varphi,\tilde\psi) = \sup_{\...
Giovanni Arquimedes Wences Naj's user avatar
2 votes
1 answer
89 views

Is the polar dual of a semi-algebraic convex body also semi-algebraic?

Call a convex body $C\subset\Bbb R^n$ semi-algebraic if it can be written as $$(*)\quad C=\bigcap_{i\in I}\, \{x\in \Bbb R^n\mid p_i(x)\le 0\}$$ with polynomials $p_i\in\Bbb R[X_1,...,X_n]$ and a ...
M. Winter's user avatar
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7 votes
1 answer
494 views

Compactness of the unit ball of a Banach space for topologies finer than the weak* topology

Let $(\mathcal{X} , \|\cdot \|_\mathcal{X})$ be a Banach space and $\mathcal{X}'$ its topological dual. We denote by $\| \cdot \|_{\mathcal{X}'}$ the dual norm and define also the topological dual $\...
Goulifet's user avatar
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8 votes
1 answer
762 views

On the Euler characteristic of a Poincaré duality space

Background. Suppose that $M$ is an oriented, connected, closed manifold of dimension $d$ with fundamental class $\mu \in H_d(M;\Bbb Z)$. Let $\Delta : M \to M \times M$ be the diagonal map. Then the ...
John Klein's user avatar
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1 vote
1 answer
147 views

Interpreting mincost flow dual variables

Consider the task of finding flow of size $b$ with minimum possible cost. It may be formulated as linear programming in a following way: $$\boxed{\begin{gather} \min\limits_{f_{ij} \in \mathbb R} &...
Oleksandr  Kulkov's user avatar
0 votes
1 answer
130 views

$f=0$ in $H^{-1}(\Omega)$ implies $f=0$ almost everywhere

Does $f=0$ in $H^{-1}(\Omega)=(H^1_0(\Omega))^*$ implies $f=0$ almost everywhere in $\Omega$?
Billal Elhamza's user avatar
1 vote
1 answer
223 views

Product-coproduct duality

Let $T$ be a set, $R$ be a ring with $1$ and $B, S_t$ be $R$-modules $\forall t \in T$ My task is to state and prove the dual to the following statement: Given momomorphisms $j_t: S_t \rightarrow B$. ...
Igor Kharin's user avatar
2 votes
1 answer
145 views

References on duality of fractional order Sobolev spaces

I would like to ask you for any good references regarding fractional order Sobolev spaces. I know Hitchhiker's guide to the fractional Sobolev spaces is a very popular one, and I found it to be quite ...
Manuel Cañizares's user avatar
1 vote
0 answers
34 views

Transport-type duality for preduals of $C^{k,1}$-functions

Let $\Omega$ be a non-empty, simply connected, and open subset of $\mathbb{R}^d$ for some positive integer $d$. Let $k$ be a non-negative integer. Consider the Banach space $C^{k,1}_0(\Omega)$ ...
ABIM's user avatar
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1 vote
0 answers
78 views

References: properties of $L_p$ spaces involving time and probability space

Questions are from the theory of PDEs\SPDEs Question 1. Suppose $(V, H, V^\star)$ is a Gelfand triple (embeddings are continuous and dense, so $\|\|_H \le C \|\|_V$ for some $C>0$ etc) of ...
just a servant's user avatar
1 vote
0 answers
167 views

Gross-Hopkins duality

$\DeclareMathOperator\Spf{Spf}\DeclareMathOperator\Pic{Pic}\DeclareMathOperator\Mod{Mod}$One can deduce the invertibility of the Gross-Hopkins dualizing spectrum from purely algebro-geometric ...
KingVon's user avatar
  • 428
4 votes
0 answers
115 views

Is there Sperner-type lemma where the colors are on the edges?

I am looking for Sperner-type lemmas where: The colors are on the edges (1-dimensional simplices) - rather than on the vertices; The target is a fully-colored vertex (a vertex adjacent to n edges of ...
Erel Segal-Halevi's user avatar
4 votes
0 answers
135 views

How does Gabriel–Ulmer duality extend to (limit, colimit) sketches?

$\newcommand\Sketch{\mathit{Sketch}}\newcommand\Set{\mathit{Set}} \DeclareMathOperator\Lim{Lim}\DeclareMathOperator\Colim{Colim}\DeclareMathOperator\Mod{Mod}\newcommand\mod{\operatorname{mod}}\...
Tim Campion's user avatar
  • 55.4k
2 votes
0 answers
106 views

Heap torsors and dual objects

Suppose that $G$ is a group and $P:G\rightarrow G$ is a permutation of $G$. We would now like to apply a forgetful functor to the algebra $(G,\cdot,P,^{-1},e)$. Whenever $g\in G$, let $gP$ be the ...
Joseph Van Name's user avatar
3 votes
0 answers
59 views

Dual space of Carleman functions

Let $X$ be the space of all weakly measurable functions $\gamma:\mathbb{R}^n \to L^2(\mathbb{R}^n)$ (modulo functions that are 0 almost everywhere) for which $$\|\gamma\|_X^2 := \sup_{\|g\|_{L^2}=1} \...
Janik's user avatar
  • 141
0 votes
0 answers
133 views

Is $(K^*)^{**}=(K^{**})^*$ for any cone $K$?

I'm considering the dual cone $K^*$ of a non-convex cone $K$. I came up with a theory that $K^{**}$ is the closure of convex hull of $K$. Then I wonder whether $(K^*)^{**}=(K^{**})^*$ holds for any ...
Dong Ganzhe's user avatar
3 votes
0 answers
216 views

Does Grothendieck duality hold without taking RHom?

I asked this very basic question about Grothendieck Duality on the Stack--exchange some time ago, without any replies. I'm therefore asking the question here to test my luck. Let $f:X\to Y$ be a ...
Adam's user avatar
  • 61
3 votes
1 answer
241 views

Confusion about the (Grothendieck–Poincaré) double dual of reflexive differentials vs usual differentials on a normal Cohen–Macaulay scheme

$\DeclareMathOperator\Hom{Hom}$Let $\mathcal{A}$ be an abelian category, my question is about the case when $\mathcal{A}$ is the category of quasi-coherent sheaves on a scheme $X$. There is a fully ...
Adam's user avatar
  • 61
5 votes
1 answer
397 views

Unifying two definitions of $L^\infty$

Let $X$ be a locally compact Hausdorff space and $\mu$ a Radon measure on $X$. Definitions: A subset $E\subseteq X$ is called locally Borel if $F \cap E$ is Borel for every Borel set $F\subseteq X$ ...
Andromeda's user avatar
4 votes
0 answers
134 views

Is the Serre dualizing complex local in the analytic topology?

There is a Serre dualizing complex $S_X\in D^b Coh(X)$ for any scheme $X$ of finite type. For proper schemes, it is characterized as the sheaf defining the right adjoint to derived global sections, ...
Dmitry Vaintrob's user avatar
8 votes
1 answer
1k views

Why aren‘t op and co switched?

When reading through Loregian and Riehl - Categorical notions of fibration, on p. 3 there is a remark that confuses me about notation. Given a $2$-category $\mathcal C$ one usually defines $\mathcal C^...
Bixxli's user avatar
  • 291
4 votes
0 answers
214 views

Exactness of a term after taking Pontryagin dual: a step in the proof of Poitou-Tate duality

I'm reading the proof of Poitou-Tate duality in the book Galois Cohomology and Class Field Theory by David Harari. After some arguments, we get a exact sequence $$ \mathbf{P}^1_S(k,M^{'})^* \...
Mugenen's user avatar
  • 141
12 votes
2 answers
357 views

Reference for free symmetric monoidal categories with duals on symmetric monoidal categories

The inclusion $U$ of the (2,1)-category of symmetric monoidal categories with duals into the (2,1)-category of symmetric monoidal categories admits a left 2-adjoint functor $L$ for formal reasons. In ...
Dmitri Pavlov's user avatar
1 vote
0 answers
501 views

Does weak-* convergence in $W^{1,\infty}$ imply weak-* convergence in $L^\infty$?

Let $\Omega \subset \mathbb{R}^n$ be open and bounded. What does weak-* convergence for a sequence of functions $\{f_k\}_{k \in \mathbb{N}}$ in $W^{1,\infty}(\Omega)$ mean? It seems to me that there ...
vampip's user avatar
  • 13
4 votes
1 answer
102 views

Order of factors for unit/counit of duals

While working through Categorical quantum mechanics by Abramsky and Coecke, I noticed that their definition of the order of factors for the unit and counit of a dualizable object disagree with the one ...
Alec Rhea's user avatar
  • 8,294
1 vote
0 answers
133 views

$L_p$ estimate in mixed boundary problem for elliptic equation

Let $Q$ be convex polygon, $\Gamma$ be a portion of boundary $\partial Q$ and $H^1_\Gamma(Q)=\lbrace u\in H^1(Q): u|_\Gamma=0\rbrace$. For $f\in (L_2(Q))^2$ consider the problem $$ \int_Q A(x)\nabla u ...
user1899's user avatar
1 vote
1 answer
87 views

Does coercivity/supercoercivity conjugates?

According to Wikipedia, a function $f: \mathbb{R}^n \to \mathbb{R} \cup \{-\infty, +\infty\}$ is called coercive if, $$f(x) \to +\infty \text{ as } \|x\| \to +\infty$$ and it is super-coercive if $$\...
Norman's user avatar
  • 125
1 vote
1 answer
84 views

Self-dual hypergraph on $\omega$

Let $H=(V,E)$ be a hypergraph. For $v\in V$ we let $v^* = \{e\in E:v\in e\}$. We define the dual of $H$ by $H^*= (E, V^*)$ where $V^* = \{v^*: v\in V\}$. We say that a $H$ is self-dual if $H \cong H^*$...
Dominic van der Zypen's user avatar
10 votes
0 answers
553 views

Reference request for a complete and formal Duality Principle in category theory

Most textbooks on category theory only sketch the meaning of the Duality Principle. But even when they do it more formally, I have only seen a version so far which concerns the language of a (single) ...
Martin Brandenburg's user avatar
5 votes
1 answer
262 views

Does rigidity imply a unique dualizing functor?

Let $\mathcal{C}$ be a rigid, monoidal category. Can I talk about $\mathcal{C}$ as having a unique, well-defined, dualizing functor (i.e. one that maps objects and morphisms onto their respective ...
Arthur's user avatar
  • 1,277
35 votes
2 answers
1k views

What are all the natural maps between iterated duals of vector spaces, and equations between these?

Fix a field. Given a vector space $V$ it has a dual $V^\ast$, which has its own dual $V^{\ast \ast}$, which has its own dual $V^{\ast \ast \ast}$, and so on ad infinitum. There are lots of natural ...
John Baez's user avatar
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