Questions tagged [duality]
Use for questions regarding duality of mathematical object, i.e. dual spaces, objects with two possible interpretations etc.
312 questions
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Duality of Bochner $L^{\infty}$ space
Let's have a look to the unit interval $[0,1]$ and a Banach space $X$ and then to the space
$$
E:=L^{\infty}([0,1],X),
$$
i.e. all essentially bounded Banach-valued functions $f:[0,1]\rightarrow X$. ...
2
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0
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158
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The dual of the space of continuous sections in a vector bundle
If $X$ is a compact Hausdorff space, one may view the space of complex, continuous functions on it as the space of continuous sections in the trivial Hermitian bundle $X \times \mathbb C$. By the ...
- 5,407
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2
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716
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Is the realtive dualizing sheaf Cohen-Macaulay?
Let $k$ be an algebraically closed field and let $X$ be a finite type $k$-scheme that is Cohen-Macaulay and equidimensional. Under these assumptions there is a relative dualizing sheaf $\omega_{X/k}$ ...
- 825
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49
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Weakly reflexive algebra vs proper (residually finite-dimensional) algebra
Currently I am reading the book "Hopf Algebras. An Introduction" by S. Dascalescu, C. Nastasescu, S. Raianu. There is a Definition 1.5.20 on page 44 (boldface is mine):
An algebra $A$ is called ...
- 519
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3
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dual space of a subspace of the space of bounded measures
Let $\mathcal{M}=\mathcal{M}(\mathbb{R})$ be the space of bounded measures. Equipped with the weak convergence, the dual space of $\mathcal{M}$ is $\mathcal{C}_b(\mathbb{R})$ consisting of continuous ...
- 1,835
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606
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Dualizing sheaf on a Cohen-Macaulay scheme
I would like to fill a detail in the answer given by Francesco Polizzi to the question "Is the dualizing sheaf on a Cohen-Macaulay scheme reflexive?".
Let $X$ be a normal, Cohen-Macaulay scheme of ...
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59
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Nests on Banach spaces and their duals
Let $X$ be a Banach space and $\mathcal{E}$ a nest on $X$.
Take $f\in X^{*}$ and suppose:
$N \in\mathcal{E}$ is the largest element of the nest so that $f \in N^\bot$
$N=\bigcap_{M>N}M$
Is there ...
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Which categories are the categories of models of a Lawvere theory?
Background: a Lawvere theory $T$ is a category with finite products such that each object is a power of a fixed object $x$. Given a Lawvere theory $T$, the category $\text{Mod}_T$ of models of $T$ is ...
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Grothendieck duality for resolution of singularities
I would like to know a reference for Grothendieck duality in a resolution of singularities. More precisely, let $Y$ be a normal, Gorenstein variety with finite quotient singularities, and suppose that ...
- 577
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141
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Langlands dual and integrable representations
Assume I successfully classified the integrable representations of a certain semi-simple Lie group $G$. Given this information, what do I know about the integrable representations of $G^\vee$, the ...
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1
answer
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For univariate polynomial is non-negativity on an interval equivalent to having a nonnegative scalar product with non-negative polynomials
Let $\mathbb R_d[t]$ be the set of univariate polynomials in the variable $t$ of degree $d$, and $S$ be the set of elements of $\mathbb R_d[t]$ that are nonnegative on $[0, 1]$. Does the following ...
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146
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Is there any dual relationship between quasi-completeness and barrelledness?
In the theory of stereotype spaces, it is known that for a locally convex space $X$,
If $X$ is pseudocomplete, then $X^{\star}$ is pseudosaturated, and
If $X$ is pseudosaturated, then $X^{\star}$ ...
- 419
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3
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Do any Stone-like dualities have some self-dualities hidden inside them?
This question originated from the observation that in most cases when one has duality of structured sets induced by a dualizing set-with-two-structures $D$, both sides of the duality are substructures ...
- 18.4k
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Legendre transform on signed measure space
Let $X$ be an open set in $\mathbb{R}^n$ and $M(X)$ be the space of finite signed measures defined on $X$. $L(p)$ is a lower-semicontinuous convex functional defined on $M(X)$. My question is: (1) ...
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Topologies of pairs and closed bounded convex sets
[I have posted this question on MSE some time ago, but received no answer.]
It is known, that if two locally convex topologies on a vector space determine the same collection of continuous linear ...
- 5,529
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Dual of Banach-valued $L^p$ [duplicate]
Let $X$ be an infinite-dimensional Banach space and let $p\in(1,+\infty)$. We may define $L^p(\mathbb R;X)$. Is it always true that the topological dual of $L^p(\mathbb R;X)$ is $L^{p'}(\mathbb R;X^*)...
- 16.2k
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48
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Nondegenerate linear maps functorially associated to algebras
In the sequel, "$k$-algebra" means "associative unital finite dimensional $k$-algebra.
Apologies for the very long exposition.
If $A$ is a $k$-algebra and $s:A\to k$ is $k$-linear, we say that $s$ ...
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Existence of dominating measure for weak*-compact set of measures
I have posted the following question also here a longer time ago, but due to no answers I thought it might fit better to MO.
Let $(\Omega,\mathcal F)$ be a measurable space and $\mathcal P$ a weak*-...
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Dual of colimit in $\text{Ban}_1$
I learned in J. Castillo's Hitchhiker guide to categorical Banach space theory that, by a theorem of Semadeni and Zidenberg, limits and colimits exist in the category $\text{Ban}_1$ of Banach spaces ...
- 393
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1
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235
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Criterion for weak compactness
Let $F$ be a metrizable locally convex space (you may assume it is a Banach space), and let $E$ be a complete locally convex space (you may assume it is a Frechet space). Let $T$ be a continuous ...
- 5,529
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How do I find the Rockafellar-Fenchel dual derivation of reguarlized objective function?
I'm stuck finding the dual of this optimization problem:
$$ \max_{\tilde{u},\tilde{v}} \int \left( \frac{1}{2}\left\|x\right\|^2 + \tilde{u}(x)\right) \, ds(x) + \int \left( \frac{1}{2}\left\|y\right\|...
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Arithmetic projective duality
Projective duality is a duality that associates to a (smooth) subvariety X of $\mathbb{P}^n$ the dual variety $X^*\subset\mathbb{P}^{n*}$ of tangent hyperplanes.
What makes the duality interesting ...
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215
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Dual cone of 'positive' Bochner integrable functions
If we consider the space of integrable functions $L^1([0,1];\mathbb{R})$, it can be ordered by the convex cone of positive integrable functions $L^1([0,1];\mathbb{R}_+)$. It is known that the ...
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Relative dualizing sheaf (reference, behavior)
Let $\mathcal{C}\rightarrow S$ a flat projective family of locally complete intersection projective curves over a integral noetherian scheme (say a spectrum of a local ring). I was wondering whether ...
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Equivalent Norms for the Dual of Sobolev / Bessel Spaces
Using standard notation, we refer to $H^s(\mathbb R) = W^{s,2}(\mathbb R)$ to be the Sobolev Hilbert spaces. As is often the case, it's natural to then consider properties of functions in $H^s(\mathbb ...
- 289
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154
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Weak$^*$ topology on Bochner $L^p$-spaces
Assume that $X$ is a Banach space such that the dual $X'$ has the Radon-Nikodym property. Moreover let $(\Omega,\Sigma,\mu)$ a say finite measure space. Then we know that for $1\leq p<\infty$ holds ...
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Duality between large and small scale structures
A rather immediate reaction to seeing the definition of a coarse structure, at least to me, is to be reminded of a uniform structure. The axioms for a coarse structure $\mathcal{C}$ (defined by a ...
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Characterizing the Dual of $W_0^{s,p}$
I am interested in literature/results characterizing the dual of the fractional Sobolev space $W^{s,p}(\Omega)$, where $\Omega \subset \mathbb{R}^N$ is open, bounded, and smooth, $0< s<1$, and $...
- 882
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1
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225
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Is the biproduct of dualizable objects itself dualizable
In a monoidal category with biproducts, let $A$ and $B$ be objects with right duals. Then does $A \oplus B$ have a right dual?
The question is a bit subtle. Suppose I already know that $A \oplus B$ ...
- 2,513
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952
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Dimensions of dual vector spaces
Let $V_F$ be an infinite dimensional right $F$-vector space (over a field $F$, or even over a division ring). The dual space $V^{\ast}={\rm Hom}(V,F)$ is naturally a left $F$-vector space (coming ...
- 18.7k
3
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2
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Why is the norm map dual to restriction under Tate local duality?
Let $L/K$ be a finite Galois extension of nonarchimedean local fields, and let $A$ and $A^t$ be dual abelian varieties over $K$. Tate local duality tells us that $A^t(K)$ and $H^1(K, A)$ are ...
- 1,103
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Matlis' dual of injective modules
Let $(R, \mathfrak{m})$ be a commutative Noetherian complete local rings ($R$ can be regular, if you need). Let $E(R/\mathfrak{p})$ be injective hull of $R/\mathfrak{p}$, if $\mathfrak{p}= \mathfrak{m}...
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How do you compute the dual norm of an induced norm on a subspace of a finite-dimensional $\ell^p$-normed vector space?
Say you have a finite-dimensional vector space $V$ with an $\ell^p$ norm on it. In general, the norm induced on a subspace $V_s$ of doesn't have to be another $\ell^p$ norm, so the unit sphere in $V_s$...
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1
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Creating Duals in A Category
Before stating my question I would like to provide afew motivating examples:
Examples:
In the category of Finitely-generated-projective $R$-modules, we have that:
$M^{\vee}:=Hom_R(M,R)$ satisfies: $...
- 5,405
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Dual representation of problems involving $f$-divergences
Studying some problems arising in decision-making under model uncertainty, I'm led to consider the following problems.
Let $\mathbb E_P$ and $\mathbb V_P$ denote the expectation and variance ...
- 6,853
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Discrete primal-duality in optimization
I would like to inquire about the existence of something perhaps similar to the duality theorem in the convex analysis or convex programming in the discrete setting. Here is a concrete example.
Let $...
- 2,251
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1
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771
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Why is the Tate local duality pairing compatible with the Cartier duality pairing?
This question is a follow up to Why is the norm map dual to restriction under Tate local duality?
Let $A$ and $B$ be dual abelian schemes over a base scheme $S$. For an integer $n \ge 1$, consider ...
- 1,103
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1
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Do doubly infeasible Linear Programming problems always have doubly infeasible bases?
Consider a Linear Programming problem in dictionary form,
$$\max\Big\{f^\pi+\!\!\!\sum_{j\in D(\pi)}\!\! d^\pi_jx_j~\Big|~\forall~i\!\in\!B(\pi)~~~ b^\pi_i+\!\!\!\sum_{j\in D(\pi)}\!\! G^\pi_{ij}x_j\...
- 21
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How is the duality pairing of $H^{1/2}$ and $H^{-1/2}$ defined on a subset of the boundary?
Let $\Omega\in \mathbb{R}^d$,for $d\in \{2,3\}$ be a bounded polyhedral set with $n$ boundary faces labeled $\{e_i\}_{i=1}^n$.
Let $\vec{q}\in H^{\mathrm{div}}(\Omega).$ Given $u\in H^{1/2}(\...
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Is the category of profunctors $Prof(A,B)$ equivalent to $Prof(B,A)^{op}$?
$\def\Prof{\mathsf{Prof}}\def\Set{\mathsf{Set}}\def\tobar{\mathrel{\mkern3mu \vcenter{\hbox{$\scriptscriptstyle+$}}\mkern-12mu{\to}}}$Let $A$ and $B$ be categories. Define a profunctor $A\tobar B$ to ...
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On Radon-Nikodym property of a dual of a Bochner space
Let ${\rm S}^{n-1}$ be the unit sphere of ${\bf R}^n$ and let us consider the dual of the space $L^p\left({\bf R}^n; C^1({\rm S}^{n-1})\right)$, for some $p\in\langle1,\infty\rangle$: it is the space ...
- 165
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600
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Grothendieck-Verdier duality for affine morphisms
Suppose $X,Y$ are varieties over $\mathbb{C}$, $Y$ is smooth and $X$ is Gorenstein ($X$ is not smooth in my case). Let $f: X \to Y$ be an affine morphism, and each fibre of $f$ has the same dimension $...
- 3,472
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Associator of the "dual" monoidal structure of a Grothendieck--Verdier Category
In A duality formalism in the spirit of Grothendieck and Verdier, Boyarchenko and Drinfeld consider a monoidal category $(\mathcal{M}, \otimes, \mathbf{1})$ together with an object $K \in \mathcal{M}$ ...
- 95
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Duality and Euler paths in graphs
I'm computer scientist and in one of my researches I'm facing this question:
if I have a planar graph that admits an Euler path (i.e. has 0 or 2 odd degree vertices, as Euler's theorem says), then his ...
- 21
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On the dual of Banach space [closed]
let $X$ be a (complex) Banach space, and $\{x_n\}$ is a sequence in $X$. Suppose that for any $f\in X'$, $$\sum_{n=1}^\infty |f(x_n)|<\infty.$$ Show that there exists a constant $\mu>0$ such ...
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350
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Predual of a subspace
Let $E$ be a Banach space, let $d\ge 1$ be an integer. Let $\mathcal G$ be a weakly closed subspace of
$(E^*)^d$ with finite codimension.
I would like know if the space $\mathcal G$ is a dual space $\...
- 16.2k
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1
answer
2k
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Questions about spectra of rings of continuous functions
I have been thinking a bit about rings of continuous functions of various kinds -- how they motivate the more modern notion of the Zariski topology on the prime spectrum as well as how they fit into a ...
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Dual Norm For Sum of 2-Norms
What is the dual of a norm that is the sum of two-norms? Specifically, say we have the following norm for $\mathbf{x}\in \mathbb{R}^n$ and $\mathbf{A}_i \in \mathbb{R}^{m \times n}$
$\|\mathbf{x}\| = ...
5
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0
answers
130
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Quantum subgroups of Locally Compact Groups and Parabolic Induction
In the classical theory , parabolic induction is used to construct the (reduced) dual of a (semi-simple) Lie Group. However, for this we need subgruops. Given that the theory of "quantum subgroups" of ...
- 561
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1
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594
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Is dual cone unique? [closed]
Suppose we have the following relationship, note that $A,B,C$ are closed convex matrix cones,
$A^\ast=C,$
$B^\ast=C,$
can we state that $A=B$? Is the dual cone of a cone is unique?
the definition ...
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