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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Can these characterisations of finite dimensionality be proven equivalent without using a basis?
I was wondering about how to define "finite dimensional" without talking about bases. Two possibilities occurred to me:
Say $V$ is finite dimensional if the canonical inclusion $V\hookrightarrow V^{**...
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If $f_n \overset{\star}{\rightharpoonup} f$ in $\sigma(E^\star, E)$, then $\|f\| \le \liminf \|f_n\|$
I'm trying to prove this result. Could you have a check on my proof?
Let $(E, | \cdot|)$ be a normed linear space and $E^\star$ its topological dual. Let $\sigma(E^\star, E)$ be the weak$^\star$ ...
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Continuous bilinear maps on sections of vector bundles
Let $E \to M$ and $F \to N$ be two vector bundles over smooth manifolds $M, N$. Denote by $\pi_1, \pi_2$ the projections of $M \times N$ to $M$ and $N$, respectively. Equip the spaces of sections of a ...
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If E* is separable, so is E. Is this proof correct?
This is my first time posting here. I already saw a post regarding this result, but I wanted to check if this specific proof is correct or not; as the proof presented in the book is different. Also, I'...
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$H$-comodule structure of $A\otimes_K A$
I'm studying the paper "Hopf Galois Theory: A survey" by Susan Montgomery, and need some help with something that the paper does not define (and that I'm not able to find anywere). Consider ...
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What is the Double and Triple Dual of $\ell^\infty$
Suppose that we start with $c^0$ and keep taking duals,
$$c^0 \to \ell^1 \to \ell^\infty \to \ell^{\infty*} \to \ell^{\infty **} \to \ell^{\infty***} \to \ell^{\infty****}\to \dots$$
Is this an ...
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functional derivative and dualspace
Consider the function space $F=\{ f : \mathbb{R}^m \rightarrow \mathbb{R}^n\}$ and the empirical scalarproduct:
$$
\langle f,g\rangle:=1/n\sum^n_{i=1}f(x_i)^Tg(x_i),
$$
for a a finite dataset $x_1, \...
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If $\langle f,\varphi\rangle$ is integrable for every functional $φ$, is there an $x$ with $\langle x,\varphi\rangle=\int\langle f,φ\rangle$?
Let $(\Omega,\mathcal A,\mu)$ be a measure space, $E$ be a normed space and $f:\Omega\to E$ be strongly$^1$ $\mathcal A$-measurable with $$\langle f,\varphi\rangle\in\mathcal L^1(\mu)\;\;\;\text{for ...
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Dual Space of General $L^p$ Space Which Takes Values in Banach Space
Last week, our functional analysis course covered Riesz Representation theorem for $L^p(X,\mu),(1\leq p < \infty)$, namely, $(L^p(X,\mu))^* = L^q(X,\mu)$. And I was stuck with this homework problem ...
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Exercise 3, Section 3.6, Hoffman and Kunze's *Linear Algebra*, 2nd edition
Hoffman and Kunze, Exercise 3, Chapter 3.6 is in the section called "The Double Dual" and on first reading it sounds very straightforward. But I really struggled with it, particularly to find a ...
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Exercise 11, Section 6.6 of Hoffman’s Linear Algebra
Let $V$ be a vector space, let $W_1, \ldots, W_k$ be subspaces of $V$, and let
$$V_j = W_1 + \cdots + W_{j-1} + W_{j+1} + \cdots + W_k.$$ Suppose that $V = W_1 \oplus \cdots \oplus W_k$. Prove that ...
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Generalize Theorem 1. in Diestel/Uhl's Vector Measures
I'm reading Theorem 1 at page 98 of Vector Measures by Joseph Diestel, John Jerry Uhl. Here we use the Bochner integral.
Theorem 1 Let $(\Omega, \Sigma, \mu)$ be a finite measure space, $1 \leq p<\...
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On what condition will make "$S^{\perp} \subseteq T^{\perp} \Longrightarrow T \subseteq S$" true?
Suppose that $V$ is a $\mathbb{F}$-vector space with some $\mathbb{F}$ a field, and denote $V^{'} := \{l: V \to \mathbb{F} \,\, \mid $ $l$ is linear$\}$, the dual space of $V$. Define the annihilator ...
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What can we say about eigen values of adjoint of a linear operator
$T: V \to V$ be a linear operator. Adjoint of T is defined as $T^{×}: V' \to V'$ and $T^{×}g = gT$.
If $\lambda$ is an eigen value of T such that $T(x)= \lambda x$ for some non zero vector $x\in V$. ...
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Is there a very small gap or no gap in this proof? ("Linear Algebra Done Right 3rd Edition" by Sheldon Axler.)
I am reading "Linear Algebra Done Right 3rd Edition" by Sheldon Axler.
Let $V$ be a vector space.
Let $V'$ be the dual space of $V$.
Let $W$ be a vector space.
Let $W'$ be the dual space of $...
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What's the problem with the evaluation map not being continuous?
When introducing differentiable functions between locally convex spaces, many authors (e.g. Bastiani, Keller, Kriegl-Michor) notice that the evaluation map
$$ E\times E^*\to\mathbb R,\qquad (x,L)\...
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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 the dual of $l^p$ for $0<p<1$?
$l^p$ for $1\leq p<\infty$ is defined as the vector space on which $$x\mapsto\sqrt[p]{\sum_j{|x_j|^p}}$$ is a norm. For $0<p<1$, this cannot be a norm, but (as Wiki indicates) the function $$...
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Proof that $V^k \mapsto W \cong T^{(k, 0)}V \otimes W$
I am very new to all of this, so please point out if I am misunderstanding something crucial. Anyhow, if $V^k \mapsto W$ is the space of $k$-linear maps from $V^k$ to $W$, then I am trying to prove ...
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Pre-dual of the measure space $\mathcal{M}(X)$
I have to find the 'pre-dual of the measure space $\mathcal{M}(X)$'. $X$ can be assumed to be Polish and equipped with the Borel $\sigma$ algebra. This is all I'm given and it's a bit vague. What I ...
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Prove 'dual to the dual norm is the original norm' without using Hahn-Banach theorem?
Let $\|\cdot\|$ be a norm on $\mathbb{R}^{n} .$ The associated dual norm, denoted $\|\cdot\|_{*},$ is defined as
$$
\|z\|_{*}=\sup \left\{z^{\top} x \mid\|x\| \leq 1\right\}
$$
I'm trying to prove
$$\|...
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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 space of the Intersection of locally convex vector spaces
Let $S \neq \emptyset$ and let $\big((E_s,\mathcal{T}_s)\big)_{s \in S}$ be a family of locally convex vector subspaces of the same vector space. Denote by $E_s^*$ the dual space of $(E_s,\mathcal{T}...
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algebraic and topological dual (vector space)
If we have a infinite dimensional topological vectorspace $X$, how can one show that the algebraic dual $X^*$ and the topological dual $X'$ are not the same. We thought about using the fact that for $...
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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
\begin{equation}
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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Is the Mackey topology $\tau(l^{\infty},l^{1})$ strongly Lindelöf?
Let $l^{\infty}$ (respectively, $l^{1}$) be the space of bounded
(respectively, absolutely summable) real sequences. I need to find out if
$l^{\infty}$ equipped with the Mackey topology $\tau(l^{\...
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Is any dual metrizable locally convex space a Frechet space?
The title basically says all of it.
If a normed space $F$ is a dual of a normed space $E$, then $F$ is a Banach space. I wonder if the same holds for Frechet spaces.
The strong dual $F$ of a locally ...
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Hoffman and Kunze, Linear Algebra Sec 3.5 exercise 11
$W_1$ and $W_2$ are subspaces of a finite-dimensional vector space $V$. $W^0$
is the annihilator of $W$.
(a) Prove $(W_1 + W_2)^0 = W_1^0 \cap W_2^0$.
(b) Prove $(W_1 \cap W_2)^0 = W_1^0 + W_2^0$.
...
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What is the intuition behind "dual vectors" in the context of tensor analysis?
Recently I've been self-studying tensor analysis through the book titled "An Introduction to Tensors and Group Theory for Physicists," by Nadir Jeevanjee. I found it to be very intuitive, up ...
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Dual operator of Markovian operator
I am trying to incorporate the definition of dual operator in a Markovian setting.
Say I have a Markov kernel $K:S\times S\to[0,1]$ i.e., a mapping such that
for every $A\in S$ $x\to K(x,A)$ is a ...
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Dual of $L^p_{loc}$ and weak star convergence.
I was reading a paper and stumbled across the terms
$$f^\varepsilon\to f \text{ in } L^{p}_{loc} \Bigl( [0,+\infty) , \left[ L^p_{loc}(\mathbb{R}^2) \right]^* \Bigr)$$ and $$f^\varepsilon\...
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Finite collection of linear functionals can be separated
Suppose $V$ is a finite-dimensional $k$-vector space, consider a finite collection $\{\alpha_1, \dots, \alpha_n\}\subset V^*$ of linear functionals on $V$. Then there exists a vector $v\in V$ such ...
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Dual of a $C^*$ algebra valued continuous functions on a compact Hausdorff space
Let $A$ be a unital $C^*$ -algebra and let $X$ be a compact Hausdorff space. Consider the $C^*$ - algebra $C(X,A)$, which are the $A$ valued continuous functions on $X$. This $C^*$ algebra $C(X,A)=C(X)...
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...Let $f : \mathbb C^3 \to \mathbb C$ given by $f(x, y, z) = 5x - iz$, see that $f \in (\mathbb C^3)^*$. Calculate $T^t(f)$.
Consider the linear transformation $T : \mathbb C_2 \to \mathbb C_3$ given by $T(x, y) = (2x + y, y -x, iy)$. Let $f : \mathbb C^3 \to \mathbb C$ given by $f(x, y, z) = 5x - iz$, see that $f \in (\...
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Dual Vectors in Tensor Algebra
A nice, classical, way of understanding Tensor Products is as the image of a universal mapping $\phi$. That is, given a family of vector spaces $V_1,\cdots V_k$, the pair $(\phi,V)$ is a tensor ...
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Subspaces and Mackey topology
Let $E$ be a locally convex (Hausdorff) topological vector space. It's known that if $G$ is a linear subspace of $E$, then if $E$ has weak topology, then $G$ as a subspace also has weak topology.
We ...
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Motivation for the transpose map
This post has some answers that give some intuition as to the definition of the transpose. My rudimentary (perhaps inaccurate) understanding is that for a linear transformation $T: V \to W$, we're ...
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Banach spaces isomorphic to their dual
I'm trying to find out which Banach spaces $X$ satisfy the property:
There exist $c>0$ and a continuous linear isomorphism $T:X\rightarrow X^*$ such as for any $x\in X, |<T(x),x>|\geq c \|| x ...
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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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Proof about dual space
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 ...
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Definition of certain Gateaux differentials of the norm in E. R. Lorch: "A Curvature Study of Convex Bodies in Banach Spaces"
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 ...
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