Questions tagged [dual-spaces]

The dual space of a vector space $V$ over a field $k$ is the vector space of all linear maps from $V$ into $k$.

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Dual of $\ell^p$ Direct sum

I am asked to show that the $\ell^p$-direct sum of a sequence of Banach Spaces $X_n$ is isometrically isomorphic to the $\ell^q$ direct sum of $X_n^*$ where $X_n^*$ is the dual of $X_n$ for each $n$ ...
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understanding why Wasserstein is weak

I am reading Wasserstein GAN paper and in Appendix A, it says Let $\mathcal{X} \subseteq \mathbb{R}^d$ be a compact set (such as $[0, 1]^d$ the space of images). We define Prob($\mathcal{X}$) to be ...
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What is the dual space of a signed measures with zero mean

For locally compact space $X$, Let $C_c(X)$ be the compactly supported, continuous functions on $X$ to $\mathbb{C}$. It is known theorem (Riesz-Markov-Kakutani) that the dual space of $C_c(X)$ is ...
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What is a duality argument?

I believe this is probably a question that can have a wide variety of answers, but i believe i'm still interested. The thing is, i have been in a couple pde talks, and i saw that in both of these ...
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Dual norm and angle: Does $\arccos \frac{u^T v}{\|u\|\|v\|_*}$ have a name?

Let $\|\cdot\|$ be a norm on $\mathbb{R}^n$, and $\|\cdot\|_*$ be its dual norm. For any nonzero vectors $u,v \in \mathbb{R}^n$, define A(u,v) = \arccos \frac{u^T v}{\|u\|\|v\|_*}. \...
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Given Banach / normed space $X$ find $E$ such that $E^* = X$, where $^*$ denotes dual space.

The concept of the dual space is well known and in some cases computably up to an isometric isomorphism, i.e. $c_0^* \cong \ell_1$ or $\ell_p \cong \ell_q$. What if we are given a space $X$ and want ...
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Can you have an infinite descending chain of dual spaces?

If a Banach space $X$ is not reflexive, then you have an infinite ascending chain of (continuous) dual spaces: $X’$, $X’’$, $X’’’$, etc. None of these are isomorphic to each other or to $X$. My ...
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Commutative diagrams for vector spaces, dual spaces, and adjoint of linear maps

I'm looking for a commutative diagram explaining the relationship between a vector space, $X$ the dual space, $X^*$, the double dual, $X^{**}$. I'm also looking for a diagram explaining the ...
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[Looking for Confirmation](X,G) is a duality, then (X,G1) is also one iff G1 is dense in G

Edit: I noticed, that my original post has not had any replies, therefore I wrote the problem again (together with my attemps, which are also already more detailed that 2 days ago) and structured it ...
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When does weak convergence imply strong convergence, can something be said in general?

We know (and indeed it's trivial to show) that $x_n \to x \implies x_n \rightharpoonup x$. It is a reason for the name "weak-convergence". We have several examples on this site where we have the ...
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The image of the canonical map into the double dual of a vector space

If $V$ is a vector space over some field, we have a canonical map $\phi_V:V\to V^{**}$ from $V$ to its double dual. The map $\phi_V$ depends naturally on $V$, and its scalar multiples are the only ...
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Dual space whose predual is the closed linear span of $E$ in $X^*$.

I want to prove Exercise 5 of the chapter "Weak Topology" of the book J.Diestel "Sequences and Series in Banach Spaces". "Let $X$ be a Banach Space and $E \subseteq X^*$. ...
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Cartesian monoidal star-autonomous categories

EDIT: Note that I have cross-posted the question on MathOverflow in the meantime. 1. Question Any rigid cartesian monoidal category is trivial (see here). Star-autonomity is a generalization of ...
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Let $X$ a vectorial space y let $\Gamma \subset X^{\ast}$. We will say that $\Gamma$ is total in $X$ if $f(x)=0$, $\forall f \in \Gamma$ implies that $x=0$. I have to prove that if $\Gamma$ is total ...
In the paper E. R. Lorch: A Curvature Study of Convex Bodies in Banach Spaces from 1953, the following assumptions and definitions are stated in section II (p. 107-108): Let $(B, \| \cdot \|)$ be a ...