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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On duality of a space of functions.
I was reading up some old courses I once took and noticed a similar type of reasoning between biduals of vector spaces and the Gelfand representation of commutative (unital) C$^*$-algebras. I was ...
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Show that the linear map induced by a symmetric associative bilinear form of L is an L-module homomorphism
This is about exercise 9.10 in the book "Introduction to Lie Algebras" by Karin Erdmann and Mark J. Wildon.
Suppose that $\textbf{L}$ is a Lie algebra over $\textbf{C}$ and that $\beta$ is a ...
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Why is a vector space $V$ equal to the spectrum of the polynomial ring generated by its dual basis?
I am recently reading this paper, "On the Chow Ring of a Geometric Quotient" by [Elingsrud, Stromme, 89]. There is an identity in section 3 that is a bit confusing:
$$V = Spec (k[x_1, \dots, ...
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Why "onto" instead of just "to" in this isomorphism definition?
From the Axler's book "Linear Algebra Done Right":
Assuming that V and W are finite-dimensional vector spaces.Prove that the map that takes $T\in\mathcal{L}(V,W)$ to $T'\in\mathcal{L}(W',V')...
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Matrix of $T,T'$ and description of $T'$ in terms of the dual basis.
Here is the question I am trying to understand its solution:
Let $V$ and $W$ be finite dimensional vector spaces. Suppose that $(v_1, v_2, v_3, v_4)$ is a basis of $V$ and $(w_1, w_2, w_3)$ is a basis ...
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Extreme point of the unit ball of $K(X, Y)^*$.
Let $K(X, Y)$ be the collection of all compact linear operators from $X$ to $Y$ and $K(X, Y)^*$ be the dual of $K(X, Y)$. I am interested to know the extreme points of the unit ball of $K(X, Y)^*$. In ...
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What property of $C^{\infty}(M)$ guarentees that finitely generated module self-dual?
According to Serre–Swan theorem the category of smooth vector bundles on a smooth manifold $M$ is equivalent to the category of finitely generated, projective $C^{\infty}(M)$-modules, and note that ...
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Confusion on the dual of $H^1$
Let $H^1(\Omega)$ be the Sobolev space $W^{1,2}$ defined over a domain $\Omega \subset \mathbb{R}^n$. It has been asked many times on this site, such as in this question, whether the space $H^{-1}$ is ...
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How to show a point is a weak* -weak continuous for the identity map on $X^*$ or on $X^{**}$?
I am trying to understand the Remark 3.2 mentioned in the paper titled as "On Weak*
-Extreme Points in Banach Spaces" written by S. Dutta and T. S. S. R. K. Rao (http://library.isical.ac.in:...
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Existence of a linearly independent vectors such that $f_i(v_j) = \delta_{ij}$
I am trying to work on a problem that is similar to this question: If $f_1,\dots,f_k\in V^*$ are linearly independent, then there are $v_1,\dots,v_k\in V$ such that $f_i(v_j)=\delta_{ij}$? but would ...
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If $Y$ embeds continuously into $X$, then $X'$ is separating in $Y'$?
In this answer it is implicitly proved that if $X, Y$ are Banach spaces such that $Y$ embeds continuously and densely into $X$, then $X'$ is a separating subspace of $Y'$.
However, I don't see where ...
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Dimension of Kernel of a linear functional is co-dimension 1
Let $ B = \{ v_1, v_2, \dots, v_n \} $ be a basis of a finite-dimensional vector space $V$,
and let $\{ f_1, f_2, \dots, f_n \} $ be the corresponding dual basis i.e basis of $ V^*$, such that,
$$f_i(...
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Representation of the dual of $C_0(\mathbb{N})$ [closed]
Let define
$$ C_0(\mathbb{N}) = \{ f \in l^\infty(\mathbb{N}) | \lim_{n\to \infty} f_n =0\}.$$
I should prove that $C_0(\mathbb{N})^* \cong l^1(\mathbb{N})$.
I tried defining an operator
$$ \Phi : l^1(...
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Wouldn't this corollary imply that weak convergence implies strong convergence? (Kreyszig 4.3-4)
I'm studying "Introductory Functional Analysis with Applications" book by Kreyszig.
Corollary 4.3-4 states the following:
For every $x$ in a normed space $X$ we have:
$$ \lVert x \rVert = \...
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canonical mapping, Isomorphism, why does it matter and what is the intuition behind it? [closed]
I know this is a repeated question, but I haven't been fully satisfied with the answers I’ve seen so far. Here is my question:
We know that if there exists a canonical mapping between two vector ...
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Is a bounded linear operator $T$ an isomorphism if and only if its adjoint $T'$ is an isomorphism?
Let $E,F$ be Banach spaces. The dual spaces (sets of all linear and bounded functionals) are denoted $E',F'$. Now take a linear, bounded operator $T:E \to F$. Then $T':F' \to E', x' \mapsto x'\circ T$ ...
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A kind of Converse for "Annihilator as the intersection of $n-d$ linear functionals' kernels"
Suppose $W \subset V$ is a $d$-dimensional subspace of $V$ which is $n$ dimensional. Let $\{\alpha_1,\alpha_2 \cdots \alpha_d \}$ be a basis for $W$, and $\{\alpha_1,\alpha_2 \cdots \alpha_d, \alpha_{...
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Transposition operator is continuous? (topology of bounded convergence)
Given a continuous map of TVS $u:E\to F$, we can associate to it the transpose map
$u^t:F'\to E'$. Here, $E',F'$ are the continuous duals of $E,F$, respectively. The map $u^t$ is continuous, so long ...
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Is this relative interior defined dually always non-empty for compact convex sets?
This is a refinment of the question asked in this post. I repeat part of the setup so that it is self contained, the main difference is that I am interested in knowing if it holds for compact sets ...
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Quadratic functions between two-dimensional vector spaces (over $\mathbb R$)
I am to define the notion of a function $F:V \to W$ being quadratic using coordinate systems, show that this definition is independent of the choice of coordinates. Next, I must show that if, for any $...
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Why does the weak$^*$ topology $\sigma(E^*,E)$ agree with the one induced by the completion of $E$?
Given a normed vector space $E$, its dual space $E^*$ has a standard norm itself. However, we can define the weak$^*$ topology $\sigma(E^*,E)$ over $E^*$ to be the minimal one in which the evaluation (...
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Why do the duals maps have the same eigenvalue as the original linear map.
Suppose $V$ is finite-dimensional, $T \in L(V)$, and $\lambda \in \mathbf{F}$.
Show that $\lambda$ is an eigenvalue of $T$ if and only if $\lambda$ is an eigenvalue of the dual operator $T' \in L(V')$....
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Question About Dense Subspace of $L^p(X,\mathscr{A},\mu)$ - Proof of Theorem 4.5.1 from Measure Theory by Donald Cohn
Background
I am self-studying Donald Cohn's Measure Theory second edition. I got stuck on a step of the proof of Theorem 4.5.1. Here is the theorem:
Theorem$\quad$ Let $(X,\mathscr{A},\mu)$ be a ...
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Question About Continuous and Linear Function on $L^p(X,\mathscr{A},\mu)$ - Theorem 4.5.1 from Measure Theory by Donald Cohn
Background
I am self-studying Donald Cohn's Measure Theory second edition. I got stuck on a step of the proof of Theorem 4.5.1. Here is the theorem:
Theorem$\quad$ Let $(X,\mathscr{A},\mu)$ be a ...
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Consequence of Banach Alaoglu Theorem
The Banach Alaoglu theorem states:
Let $\mathcal{Z}$ be a banach space. The closed unit ball
$\{Z^{\ast}\in\mathcal{Z}^{\ast}:\|Z^{\ast}\|_{\ast}\leq1\}$ is compact in the weak$\ast$ topology of $\...
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Weak* compactness of subset of $L^q$
Let $\mathcal{Z}=L^p(\Omega, \mathcal{A}, \mu)$ and its dual space $\mathcal{Z}^*=L^q(\Omega, \mathcal{A}, \mu)$. Let
$$\langle Z, Z^*\rangle = \int Z \, Z^* d\mu$$
be their dual paar.
EDIT: Let $c:\...
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Is the cartier duality of a group scheme the Hom_{R-Mod}(-,R) or Hom_{R-Mod}(-,R)?
I'm reading and reassuming R. Pink "finite group schemes" lecture course notes (in particular sections 2, 3, 4, 5, 11, 12, 13, 14) as an assignment.
At section \s4 there's a definition of ...
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Proof that the Dual Transformation of a Dual Transformation is Itself
Let $A$ be an invertible linear transformation in $\mathbb{R}^{n\times n}$. Then, we know that $A$ takes $\mathbb{R}^{n-1}$ hyperplanes in its domain to $\mathbb{R}^{n-1}$ hyperplanes in its image. ...
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Is this relative interior defined dually always non-empty for closed and convex sets?
I am interested in defining a different relative interior for a closed convex set $C$ in a, possibly infinite, dimensional TVS $X$ (locally convex and Hausdorff) with topological dual $X^\ast$.
If we ...
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Why is transpose not equal to inverse in general?
Let $\phi: X \rightarrow Y$ be a linear map between finite dimensional real vector spaces, let $p:X \rightarrow X^*$ and $q: Y \rightarrow Y^*$ map from basis to dual basis, and let $\varphi: Y^* \...
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A dual space to a space of homogeneous polynomials
I stumbled across the following issue. Let say we have an $n$-dimensional vector space $V\simeq \mathbb{F}^{n}$ and let $V^{*}$ be its dual. There is no natural way to construct a natural isomorphism ...
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$T$ compact operator, Let $\Delta^*_{\bar{\lambda}}$ subspace of $X^*$. Prove $\Delta^*_{\bar{\lambda}} = \bar{\Delta^*_{\bar{\lambda}}}$
We have proved the following claim:
Let T be compact operator and $\lambda \neq 0$. Then $\Delta_\lambda = \bar{\Delta _\lambda}$.
Now there is the corollary:
$T$ compact operator, Let $\Delta^*_{\...
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How do we know the dual pairing between Lp spaces is well defined? [closed]
Let $(\Omega, \mathcal{A}, \mu)$ be a measure space and let $X \in L^p(\Omega, \mathcal{A}, \mu)$ and $Y\in L^q(\Omega, \mathcal{A}, \mu)$. Then the dual pair betweent these spaces is defined as $\...
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Understanding spaces of negative regularity
Let $C^k(\mathbb{R}^n$) be the space of functions with $k$ continuous derivatives, and $H^s(\mathbb{R}^n)$ the Sobolev space $W^{2,s}$. Their dual spaces are commonly denoted as $C^{-k}$ or $H^{-s}$. ...
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Basis and dual basis in finite dimensional vector spaces
In finite dimension, a basis $\{f_i\}\in V$ and a dual basis $\{g_i\}\in V^*$ should satisfies $\langle f_i, g_j \rangle = \delta_{ij}$.
I am wondering what is the terminology, if now I have a basis $(...
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Loomis and Sternberg Chapter 2, problem 2.19: defining the degree of a polynomial on a vector space (over R)
Exercise 2.19 of chapter 2 of L&S is:
A polynomial on a vector space V is a real-valued function on V which can be
represented as a finite sum of finite products of linear functionals. Define the ...
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Dual basis of a matrix space
I am considering the following basis $B$ of $S_2$:
$$B=\left\{\begin{pmatrix}1&0\\0&0\end{pmatrix},\begin{pmatrix}0&1\\1&0\end{pmatrix},\begin{pmatrix}0&0\\0&1\end{pmatrix}\...
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If $f : U \rightarrow X$ is an isometric inclusion of Banach spaces, does $f' : X' \rightarrow U'$ have a bounded generalized inverse?
Let $X$ be a Banach space and let $U$ be a closed Banach subspace. The inclusion mapping $$ f : U \rightarrow X $$
induces a dual mapping
$$ f' : X' \rightarrow U' $$
I am wondering about the ...
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If $ \phi: X \to X^* $ is an isometry, then $ X $ is a complete space.
I am wondering if the following statement might hold (as I wanted to use this in solving another problem): If $ \phi: X \to X^* $ is an isometry, then $ X $ is a complete space. I know that $ X^* $ is ...
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I want to prove that $T$ belongs to $B(X)$, meaning it is a bounded linear operator.
Let $T_f: X \rightarrow X$ be defined by $T_f(x) = f(x) u$ for every $f \in X^*$ for some non-zero $u \in X$. I want to prove that $T$ belongs to $B(X)$, meaning it is a bounded linear operator.
I ...
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Origin of notation of a dual space
Classically, there’s discordance in the notation of the dual of a vector space over a field $K$. Of course we can all agree that the algebraic dual is the vector space of linear maps from $V$ to its ...
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Proving a linear functional belongs to $L^{4/3}(0,T; H^{-1})$
Let $H^{-1}$ be the dual to the Sobolev space $H^1$ and $T > 0$. Let $u: \mathbb{R}^3 \rightarrow \mathbb{R}^3$ such that $u \in L^\infty(0,T; L^2) \cap L^2(0,T; H^1)$. It can be shown that $$(u \...
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why quojections are prequojections [closed]
While reading about quojections and prequojections in the book "Advances in the Theory of Fréchet Spaces," I'm having trouble understanding why every quojection is necessarily a ...
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Prove that the canonical mapping from an infinite dimensional vector space to it's double dual is a one-to-one mapping. [closed]
What is the canonical correspondence from a vector space V to it's double dual $V^{**}.$ Prove that this correspondence is one-one.($V$ need not be finite dimensional)
I tried solving the problem in ...
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Weak Star Convergence of Integral Averages
Suppose $U$ is a Banach space and let $V\subseteq U$ be bounded, convex and (norm-)closed. Consider the Bochner-Lebesgue space $L^r(0,T;U)$ with $T>0$ and $r\in[1,\infty]$ consisting of strongly ...
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Spanning set of support functionals in dual space
I am currently studying about supporting hyperplane (or, support functional) in dual space. Since, I am new in these topics I met with the following queries:
Let $X$ be a normed space and $X^*$ be the ...
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(Copy) Set of linear functionals span the dual space iff intersection of their kernels is {0} .
I have fully understood the following question and got a motivation from it.
Set of linear functionals span the dual space iff intersection of their kernels is $\{0\}$.
My question is what will be the ...
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Let $A : X \to X$ be a linear operator such that $T_f A \in \mathcal{B}(X)$ for every $f \in X^*$. Show that $A \in \mathcal{B}(X)$.
Let $X$ be a Banach space and let $0 \neq u \in X$.
(a) For every $f \in X^*$, let $T_f : X \to X$ be an operator defined by the prescription $T_f x = f(x) u$. Show that $T_f \in \mathcal{B}(X)$.
(b) ...
user1316777
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Assume that for every $f\in X^*$, there exists $y \in X$ such that $f(x)=\langle x, y \rangle$ for every $x \in X$. Show that $X$ is a complete space.
Let $(X, \langle \cdot, \cdot \rangle)$ be a real or complex vector space with an inner product. Assume that for every $f \in X^*$, there exists $y \in X$ such that $f(x) = \langle x, y \rangle$ for ...
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Let $X$ be a reflexive space and let $f \in X^*$, $ Px = x - \frac{f(x)}{\|f\|} y, \quad x \in X. $
Let $X$ be a reflexive space and let $f \in X^*$.
(a) Show that there exists $y \in X$ such that $x - \frac{f(x)}{\|f\|} y \in \ker f$ for every $x \in X$.
(b) Let $P : X \to X$ be the mapping defined ...
user1316777