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 $k$.
1,468 questions
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Proof of $\dim U^0=\dim V−\dim U$
This is an "exercise" from Linear Algebra Done Right. I'm trying to prove the following theorem using the approach outlined by the author:
(3.125). Suppose $V$ is a finite-dimensional ...
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Riesz Representation Theorem (Rudin RCA Theorem 6.19) - Construction of the positive linear functional
This question is from the proof of the Riesz Representation Theorem (Rudin RCA Theorem 6.19) regarding the construction of the positive linear functional. The requisite positive linear functional is ...
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Proof that $\mathcal{H}_1^\star \cong \mathcal{H}_{-1}$ for Hilbert space scales associated to power of unbounded positive self-adjoint operators.
Let $N \geq I$ be a self-adjoint densely defined operator on a complex Hilbert space $(H, \langle \cdot , \cdot \rangle, \| \cdot \|)$.
For $u \in D(N^{\frac{1}{2}}) \subset H$ define $\|u\|_{1} = \|...
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Textbooks on multilinear algebra and contractions
A friend of mine is currently writing his Bachelor's thesis on the topic of elastic materials. In particular, this involves higher-order derivatives. These are naturally expressed in the language of ...
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Dual space and dual of a linear operator: simplifying and understanding the problem
Let $V$ be a finite-dimensional space over $F$, and $V^*=\mbox{Hom}(V,F)$. There is a canonical (non-degenerate) pairing
$$\langle -,-\rangle:V^*\times V\rightarrow F, \hskip1cm
\langle \phi,v \...
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When is the max set of a Priestley space closed?
A Priestley space is a compact partially ordered topological space $P$ with the property that if $x,y \in P$ and $x$ is not less than or equal to $y$, then there is a clopen upset containing $x$ and ...
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Characterization of $H^{-1}(\mathbb{R}^d)$ by Fourier transform
$H^{-1}(\mathbb{R}^d)$ is the dual space of $H^1(\mathbb{R}^d)$ which in terms of Fourier series is given by
$$H^{-1}(\mathbb{R}^d) = \{ f \in \mathcal{S}'(\mathbb{R}^d) : (1+\vert y \vert^2)^{-1/2} \...
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Separation theorems for dual pair of vector spaces
Let $(X,Y,\langle\cdot,\cdot\rangle)$ be a dual pair of real vector spaces, i.e., $\langle\cdot,\cdot\rangle$ is a bilinear form on $X\times Y$ and $\langle\cdot,\cdot\rangle$ is non-degenerate in the ...
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What is the standard name and notation for the "orthogonal" submodule in a very general setting?
As a first year undergrad student, I remember my maths teacher having us work as an exercise on the following notion of linear algebra. Let $k$ be any field, let $V$ be a $k$-vector space and let $W \...
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Why does $H^{-1}(\mathbb{R}^d)$ have different representation for $d\geq 3$ and $d<3$? [closed]
In some lecture notes the dual space of $H^1(\mathbb{R}^d)$ is denoted by $H^{-1}(\mathbb{R}^d)$ and stated that for $d\geq 3$
$$H^{-1}(\mathbb{R}^d)=\{\operatorname{div}(F):F\in L^2(\mathbb{R}^d;\...
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Replacing integral with duality pairing in Sobolev spaces
Let $\Omega$ be a bounded domain with smooth boundary. Consider $v \in H^2_0(\Omega)$ and $f \in H^{-2}(\Omega)$. In the context of elliptic PDEs, I often see the duality pairing $\langle f, v \rangle$...
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Evaluation morphism of (Fd)Vect?
What is the evaluation morphism of $\mathbf{FdVect}_\mathbb{K}$? I ask because an evaluation morphism is defined as $\varepsilon: V^* \otimes V \to 1$, but a lot of sources claim that for $\mathbf{...
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When $\mu$ and $|\mu|$ have same total mass, does it follow that $\mu$ is necessarily $\ge 0$
Suppose X is a locally compact top. space and $\mu$ a complex Radon / regular bounded measure (some will say finite instead - anyway with traditional measure theory complex measures are always like ...
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Justifying the embeddings $V\subset H \subset V'$ in Sobolev space settings
I am reading the book Infinite-Dimensional Dynamical Systems in Mechanics and Physics written by Roger Temam. And I am trying to understand how to rigorously show that
$ H \subset V' $, where $ H = L^...
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Existence of a nonzero bounded linear functional on $L^\infty(\mathbb{R})$ that vanishes on $C_b(\mathbb{R})$
Let $C_b(\mathbb{R})$ be the set of bounded continuous function on $\mathbb{R}$. I would like to show that there exists a nonzero bounded linear functional on $L^\infty(\mathbb{R})$ that vanishes on $...
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What needs to be shown to prove that the dual space of $\ell_1$ is $\ell_\infty$?
I am working on proving that $\ell_1^* = \ell_\infty$. Let $b \in \ell_\infty$ and $\Lambda$ be the linear functional on $\ell_1$
$$\Lambda(a) = \sum a_n b_n.$$
Since
$$|\Lambda(a)| \leq \sum|a_n b_n| ...
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Normal triple defined without imposing duality
In Stochastic Partial Differential Equations by Lototsky & Rozovsky (2017) it says the following:
Let $V$, $H$, and $V’$ be three Hilbert spaces. We say that $(V,H,V’)$ is a normal triple if:
$V$ ...
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$L^2$ projection in Hilbert space: is the norm given by a supremum?
Is it true that the $L^2$-norm of the projection satisfies the variational characterization below?
$$\displaystyle\|P_U(z)\|=\sup_{v\in U,\|v\|=1} \langle z,v \rangle$$
Here, $V$ is a Hilbert space ...
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Operator $R$ for residual form of PDE is a map of subset of Hilbert space to which other (dual?) space?
I was following Wolfgang Bangerth's FEM Lecture: Nonlinear problems, part 2 -- Newton's method for PDEs covering the residual form of the minimum surface equation
\begin{equation}
\begin{aligned}
R(u)...
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$V'\otimes_K W \cong \operatorname{Lin}_K(V,W)$ via the Universal Property
Let $ V $ and $ W $ be finite-dimensional vector spaces over a field $ K $, and let $ V' $ denote the dual space of $ V $. I would like to show that
$$
V' \otimes_K W \cong \operatorname{Lin}_K(V, W).
...
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If $\lambda \in \bigcap_{\delta > 0} ~\overline{\left\{ f(T) : f\in B_{L(X, Y)^*}, |f(G)| > 1 - \delta \right\}}$, $T\in L(X, Y)$ then $\lambda =$?
Let $X, Y$ be Banach spaces. Let $L(X, Y)$ be the collection of bounded linear operator from a $X$ to $Y$. We denote $B_X$ as the closed unit disk of $X$. Suppose that $G\in L(X, Y)$ such that $\|G\|=...
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Proving the dual bundle to a smooth vector bundle is also smooth
The following is exercise 11.10 in Lee's Introduction to Smooth Manifolds.
Given a smooth vector bundle $E \to M$ on a smooth manifold $M$, we want to prove the dual bundle $E^* \to M$ is also a ...
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Szekeres problem 3.19: interpretation of $O' = O$ for zero maps
In Peter Szekeres's "A Course in Modern Mathematical Physics", problem 3.19 starts by asking:
If $A : V \to V$ is a linear operator, define its transpose to be the linear map $A' : V^* \to ...
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Szekeres problem 3.19: proving a linear map is unique
In Peter Szekeres's "A Course in Modern Mathematical Physics", problem 3.19 starts by asking:
If $A : V \to V$ is a linear operator, define its transpose to be the linear map $A' : V^* \to ...
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A contradiction for dual modules
Let $M$ be a left module over a noncommutative ring $R$ and let ${}^*M$ be the space of left $R$-module maps from $M$ to $R$ given its usual right $R$-module structure defined by $(f.r)(m) = (f(m))r$, ...
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Conjugate-Duality of $\ell^p$ Direct Sums
I am reading through a textbook that mentions if $(X_n)_{n=1}^{\infty}$ is a sequence of Banach spaces, then $\left(\bigoplus_{\ell^2} X_n\right)^{*} \cong \bigoplus_{\ell^2} X_n^{*}$ isometrically (...
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How to prove the isometry $(L^p(S;X))^*=L^q(S;X^*)$ when $S$ is an atomic measure space?
I meet a problem in Analysis in Banach Space (Volume I). At page 39 the authors want to prove the following result:
Proposition 1.3.3. Let $1<p<\infty$ and $1/p+1/q=1$, and let $(S,\mathscr{A},\...
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How to judge whether $x^{**}\in X^{**}$ is in $X$ when $X$ may not reflexive? [duplicate]
I meet this problem when reading Analysis in Banach Space (Volume I). At page 523 the authors introduce the following Krein-Smulian theorem:
Theorem B.1.12. A linear subspace of $X^*$ is weak$^*$ ...
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Show that $\{f \in E^* : \frac{1}{2}\|y\|^2 - \frac{1}{2}\|x\|^2 \geq \langle f ,y-x\rangle \quad\forall y \in E\}$ is equal to the duality map $F(x)$
Let $E$ be a normed vector space. Prove that the duality map
$$F(x) = \{f \in E^*; \|f\| = \|x\| \text{ and } \langle f, x\rangle = \|x\|^2\}$$
is equal to
$$\tilde{F}(x) = \{f \in E^* : \frac{1}{2}\|...
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What justifies taking the limit out of a duality pairing? [closed]
Let $E$ be a normed vector space and $E^*$ the dual space. For any $x \in E$ and $f_n \in E^*$ what justifies writing?
$$\langle \lim_{n \rightarrow \infty} f_n, x\rangle = \lim_{n \rightarrow \infty} ...
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Existence of unique functional $h\in M^\perp$ such that $||g-h||=d(g,M^\perp)$.
Let $X$ be a normed vector space and $M$ a linear subspace of $X$. Using the Hahn-Banach Theorem, I managed to prove that
For each $g\in X^*$ there exists $h\in M^\perp$ such that $||g-h||=d(g,M^\...
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Predual of $B(X)$
Initially I was only interested in Hilbert spaces as I was learning about the trace class operators which provide a (the) predual of $B(H)$ but then I was wondering if the space $B(X)$ has a(n ...
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Intuition about these two corollaries functional analysis
These are two corollaries in functional analysis in Brezis' book. Let $E$ be a normed linear space.
For every $x_0 \in E$ there exists $f_0 \in E^*$ such that $\|f_0\| = \|x_0\|$ and $\langle f_0, ...
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Difference between measure and distribution?
A distribution on $\mathbb{R}$ is a continuous linear form on the semi-normed space $C^\infty_C$ of smooth test functions on $\mathbb{R}$ with compact support.
Does any distribution $T$ on $\mathbb{R}$...
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Prove that if $\operatorname{Ann}(W_{1}) = \operatorname{Ann}(W_{2})$. then $W_{1} = W_{2}$.
Let $S$ be any subset of $V^*$ for some finite dimensional space $V$. Define $\def\Ann{\operatorname{Ann}}
\Ann(S) = \{v \in V \mid f(v) = 0 \text{ for all } f \in S\}$. Let $W_{1}$ and $W_{2}$ be ...
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Prove that $\text{Ann}(W_{1}+W_{2}) = \text{Ann}(W_{1}) \cap \text{Ann}(W_{2})$
Let $S$ be any subset of $V^*$ for some finite dimensional space $V$. Define $\text{Ann}(S) = \{v \in V \mid f(v) = 0 \text{ for all } f \in S\}$. Let $W_{1}$ and $W_{2}$ be subspaces of $V^*$. Prove ...
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Dual of the fractional Sobolev space [duplicate]
I wonder if the dual of the fractional Sobolev space $W^{s,p}(\Omega)$ is analogous to the usual Sobolev space $W^{1,p}(\Omega)$? Here $0<s<1, 1<p<\infty$ and $\Omega$ is a bounded domain ...
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My bijection $V \cong V^*$ isn't an isomorphism?
I've been thinking really hard about the dual space recently, hoping to think about it in terms of codim 1 subspaces and seeing if there was a reasonable way to define scalar multiplication and ...
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Understanding the Dual Space of linear transformations on a Banach space
This introduction on Open Quantum systems, https://arxiv.org/pdf/1104.5242, has the following line (on page 4):
"Now we focus our attention onto possible linear transformations on a Banach space, ...
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Dual of a MIMO control system
For a SISO (single input single output) LTI (linear time invariant) system, there is a well defined notion of dual representations of the system. This duality is delt with in this question. To briefly ...
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Adjoint of an operator
Given a Hilbert space H, I know that there exists an one to one map between the space $\mathcal{B}(H)$ (continuous lineare operator from $H$ to itself) and the set of all sesquilinear form on $H$, ...
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Bidual map attaining values with smaller norm of input than original
I am working on a problem that asks for an example of $T: X \to Y$ a continuous linear map between Banach spaces so that $\exists y \in T(X)$ with $$\inf_{\,\,\,\,z \in X^{**} \\ T^{**}(z) = y } \| z \...
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Operator norms of evaluation functionals on subspace of dual space
Suppose that $X$ is a normed vector space and that $Y$ is a vector subspace of $X^*$ that separates the points of $X$.
Is it necessarily true that $\|x\| = \sup_{f \in B_Y} |f(x)|$ $\forall x \in X$?...
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Which properties has $\mu\otimes \mu$ if $\mu\in H^{-1}$?
Assume that $\mu$ is a finite compactly supported measure (say compactly supported in some set $\Omega$) in $\mathbb{R}^3$ which lies in $H^{-1}$, i.e.
$$\exists C>0:~\forall \phi \in H_0^1(\Omega):...
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Diagonalizability of transpose (dual) operator for infinite-dimension vector spaces (purely algebraic version)
I'm writing some notes on Linear Algebra and thought of the following question:
What is the relation between diagonalizability of a linear operator and its dual.
Here are my definitions: A linear ...
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Difficulty with the definition of the $W^*$-tensor product
Let $A,B$ be $W^*$-algebras, $I,J$ essential ideals of $A,B$ resp. Here we use essential ideal in the sense of this question. It is not hard to see that $I\otimes J$ is essential in $A\otimes B,$ but ...
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Implication of non-equivalent representations of states on operator algebras on the dual space
Given two states $\omega_1, \omega_2$ on the $C^*$-algebra $\mathcal{A}$, they are said non-equivalent if their GNS representations are not equivalent, i.e. if there is no $*$-isomorphism $\gamma$ ...
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set theory and dual basis
I recently became interested in ZFC set theory, and after studying a bit about it, I decided to reformulate for myself some concepts I had studied about algebra. My question is about linear ...
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Motivation behind weak convergence in Functional Analysis
Let $X$ be a normed linear space, $(x_{n})$ be a sequence in $X$, $l\in X$. $(x_{n})$ converges weakly to $l$ if for every bounded linear functional $x^{*}$ on $X$ the sequence $x^{*}(x_{n})$ ...
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Clarification on a problem about $H^{-1}$ norm of integral operator
I have this problem.
Let $f \in H^{-1}((0,1))$ be such that
$$
f(u) = \int_0^1 u(x) \,\Bbb dx.
$$
We seek to compute $N := \|f\|_{H^{-1}((0,1))}$. Show that $1/N$ is equal to the infimum of
$\|u\|_{H^...
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