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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Why do we care about dual spaces?
When I first took linear algebra, we never learned about dual spaces. Today in lecture we discussed them and I understand what they are, but I don't really understand why we want to study them within ...
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Why are vector spaces not isomorphic to their duals?
Assuming the axiom of choice, set $\mathbb F$ to be some field (we can assume it has characteristics $0$).
I was told, by more than one person, that if $\kappa$ is an infinite cardinal then the ...
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In categorical terms, why is there no canonical isomorphism from a finite dimensional vector space to its dual?
I've read in several places that one motivation for category theory was to be able to give precise meaning to statements like, "finite dimensional vector spaces are canonically isomorphic to their ...
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The Duals of $l^\infty$ and $L^{\infty}$
Can we identify the dual space of $l^\infty$ with another "natural space"? If the answer is yes, what can we say about $L^\infty$?
By the dual space I mean the space of all continuous linear ...
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Motivation to understand double dual space
I am helping my brother with Linear Algebra. I am not able to motivate him to understand what double dual space is. Is there a nice way of explaining the concept? Thanks for your advices, examples ...
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Why isn't lambda notation popular among mathematicians?
I am relatively new to the world of academical mathematics, but I have noticed that most, if not all, mathematical textbooks that I've had the chance to come across, seem completely oblivious to the ...
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Banach Spaces - How can $B,B',B'', B''', B'''',B''''',\ldots$ behave?
(ZFC)
Let $ \big\langle B,+,\cdot, \:\: \|\cdot\| \:\: \big\rangle $ be a Banach space.
Define $ \mathbf{B} \; = \;\big\langle B,+,\cdot, \:\: \|\cdot\| \:\: \big\rangle $.
Define $\: \mathbf{B}_0 =...
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The dual space of $c$ is $\ell^1$
Here is what I know/proved so far:
Let $c_0\subset\ell^\infty$ be the collection of all sequences that converge to zero. Prove that the dual space $c_0^*=\ell^1$.
$Proof$: Let $x\in c_0$ and let $y\...
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Dual of $l^\infty$ is not $l^1$
I know that the dual space of $l^\infty$ is not $l^1$, but I didn't understand the reason.
Could you give me a example of an $x \in l^1$ such that if $y \in l^\infty$, then $ f_x(y) = \sum_{k=1}^{\...
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Prove that $X^\ast$ separable implies $X$ separable
Can someone tell me if I got the following right:
Assume $X$ to be a normed vector space over $\mathbb{R}$. Prove that if the dual space $X^\ast$ is separable then $X$ is separable as well.
I'm ...
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Are there spaces "smaller" than $c_0$ whose dual is $\ell^1$?
It is well known that both the sequence spaces $c$ and $c_0$ have duals which are isometrically isomorphic to $\ell^1$. Now, $c_0$ is a subspace of $c$. My question - is there an even smaller subspace ...
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Separability of Banach Spaces
A homework problem from Folland Chapter 5, problem 5.25.
If $\mathcal{X}$ is a Banach space and $\mathcal{X}^{\star}$ is separable, then $\mathcal{X}$ is separable.
I tried the following approach: ...
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Why is a dual space a vector space?
I was wondering if some one could please shed some light on why or how a dual space itself becomes a vector space over the field. Finite-Dimensional Vector Spaces by Paul Halmos states:
. . . to ...
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$C_0(X)$ is not the dual of a complete normed space
Let $X$ be any locally compact Hausdorff space and assume that it is not compact.
I've heard that the Banach space $(C_0(X),\|\!\cdot\!\|_\infty)$ is not isometrically isomorphic to the (norm) dual of ...
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Hahn-Banach extensions from $E$ to $E^{**}$.
I was thinking the following problem while reading some functional analysis notes.
Is it possible to characterize the Hahn-Banach extensions (meaning, extensions with the same norm) of a
functional ...
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How to interpret $(V^*)^*$, the dual space of the dual space?
Suppose $V$ is a real vector space.
Then $V^*$, its dual space, is the vector space of linear maps $V\to \mathbb R$
How then do I interpret $(V^*)^*$, the dual space of the dual space?
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Finding a dual basis
This is one of my homework questions - I'm pretty sure I understand part of it.
Let $V=\Bbb R^3$, and define $f_1, f_2, f_3 \in V^*$ as follows:
$$f_1(x,y,z) = x - 2y;\quad f_2(x,y,z) = x + y + z; \...
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Prove the dual space of $l^p$ is isomorphic to $l^q$ if $\frac{1}{q}+\frac{1}{p}=1$
Prove the dual space of $\ell^p$ is isomorphic to $\ell^q$ if $\frac{1}{q}+\frac{1}{p}=1$ ($1<p<\infty$)
Define a map $J:\ell^q \to (\ell^p)'$ such that $Jy(x)=\sum_{k=1}^\infty x_ky_k,x\in \...
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Annihilator of a vector space $V$ is the zero subspace of $V^*$
I am reading Hoffman and Kunze's Linear Algebra and in Section 3.5, page 101, they define the annihilator of a subset as follows:
Definition. If $V$ is a vector space over the field $F$ and $S$ is ...
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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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Do groups have Duals?
Do groups have Duals?
Might be a bit of a simple question but it should not take too much effort to handle. Notice that I'm not saying all or automatically or anything like that.
I'm merely ...
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Is any Banach space a dual space?
Let $X$ be a Banach space. Is there always a normed vector space $Y$ such that $X$ and $Y^*$ are isometric or isomorphic as topological vector spaces (that is, there exists a linear homeomorphism ...
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What is the intuitive meaning of the dual space of a tangent space?
We know that a tangent vector is a directional derivative operartor, and the collection of all tangent vectors at a point is a tangent space. I don't understand the intuitive meaning behind the dual ...
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Dual space of $\mathcal{C}^n [a,b]$.
I just started reading a few days ago about Banach algebras using the Kaniuth's book. In this, it is said that the space $\mathcal{C}^n [a,b]$ of $n$-times continuously differentiable functions is a ...
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Iterated duals of a vector space
Let $K$ be a field and $\mathcal U$ a universe such that $K\in\mathcal U$. (Here, "universe" means "uncountable Grothendieck universe".) Let $\mathcal C$ be the category of $K$-...
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Show that the dual space of $\ell^1$ is isomorphic to $\ell^{\infty}$
I've started taking my first course in multivariable analysis and in our notes there is an exercise about the dual space of the sequence space $\ell_1$ and I'm unsure how to prove that it is ...
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Dual of $\ell_{\infty}$ is not $\ell_1$
As the title indicates I'm trying to show that $\ell_{\infty}^{*}$ is not $\ell_1$. I've shown that for p, q conjugate and finite we do indeed have $\ell_{p}^{*} = \ell_q$, with the correspondence ...
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Dual space of $c_0$
I want to prove that the dual of $c_0=\{(x_n)_{n\in\mathbb N}\subset\mathbb R : \lim x_n=0 \text{ and }\|x\|_\infty=\sup\limits_n|x_n|\}$ is $l^1$ .
So I defined the following map,
$$T:l^1\...
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If $Y\subset X$ are Banach spaces such that $Y$ is dense in $X$, is it true that $X'$ is dense in $Y'$?
If $Y$ is a dense subspace of a Banach space $(X,\|\cdot\|_1)$ and $(Y,\|\cdot\|_2)$ is a Banach space such that the inclusion from $(Y,\|\cdot\|_2)$ into $(X,\|\cdot\|_1)$ is continuous, then it is ...
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Is the dual to $C^1[0,1]$ separable?
$C^1[0,1]$ is endowed with the norm $\|f\| = \sup_{t \in [0,1]}|f| + \sup_{t \in [0,1]}|f'| $. I need to check if its dual $(C^1[0,1])^*$ is separable (I hope it is not). I am asking for the answer ...
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Expression for dual of subgaussian norm
Here is the simplest statement of my question:
Let $Y$ be a centered real random variable and define $$\|Y\|_* = \sup \left\{ \mathbb{E}[X \cdot Y] ~:~ \forall t \in \mathbb{R} ~~ \mathbb{E}[e^{tX}]...
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Intuitively Understanding Double Dual of a Vector Space
I am trying to see if someone can help me understand the isomorphism between $V$ and $V''$ a bit more intuitively.
I understand that the dual space of $V$ is the set of linear maps from $V$ to $\...
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Suppose $\forall x \in X,\sum_{n=1}^\infty|f_n(x)|<\infty$, to prove $\forall F\in X'', \sum_{n=1}^\infty |F(f_n)|\leq C\|F\|$.
Let $X$ be a Banach space, consider $\{ f_n \}_{n \ge 1} \in X'$ s.t.
$$\sum_{n = 1}^\infty | f_n(x) | < \infty, x \in X. \tag{1}\label{cond}$$
Please prove that there exists $C \ge 0$ s.t. for ...
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Can vector space dualization be made into a covariant functor?
The contravariant endofunctor $$\mathbf{Set}^{op} \rightarrow \mathbf{Set}$$ $$X \mapsto [X,2]$$
can be made into a covariant endofunctor $$\mathbf{Set} \rightarrow \mathbf{Set}$$ $$X \mapsto \...
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What is the difference between vector space and dual space?
I read that in Dirac notation, kets are elements of a vector space and bras are elements of the dual space. My question is, what is the difference between vector space and dual space, and why are bras ...
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Properties of $\{x\in X\mid f(x)=||f||\}$
Let $X$ be a normed space, $f\in X^*\setminus\{0\}$ (the continuous dual), $E:=\{x\in X\mid f(x)=\|f\|\}$. Prove that $E$ is a nonempty closed set and that $\inf \{\|x\|\mid x\in E\}=1$.
I have no ...
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Why is there no natural isomorphism between $V$ and its dual?
Question
While looking over the exercise $3.F-34$ in Linear Algebra Done Right, I encountered the following paragraph
Suppose $V$ is finite dimensional. Then $V$ and $V'$ are isomorphic, but finding ...
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Is $l_2$ on $\mathbb{R}^n$ the only norm for which it is equal to its dual norm?
Given any norm $\|.\|$ on $\mathbb{R}^n$, its dual norm $\|.\|^D$ is defined as the following:
$\|v\|^D = \sup_{\|x\|\leq 1} |(v,x)|$, where $(,)$ is the standard Euclidean Inner product. Under that ...
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Which vector spaces are algebraic dual spaces?
Let us say that a vector space $V$ is an algebraic dual space if there exist a vector space $U$ such that $V$ is isomorphic to $U^*$, the vector space of all linear maps from $U$ to the corresponding ...
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Is the topological space $ (X^{*},\sigma (X ^ {*}, X)) $ metrizable?
Definition 1: Let $X$ be a normed space. The operator
\begin{align*}
J_{X}\colon X&\to X^{**}\\
x&\mapsto J_{X}(x)(x^{*})=x^{*}(x)
\end{align*}
is called canonical lace (or canonical ...
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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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The dual of $L^p$ is $L^q$
Let $1 \leq p < \infty$ and $1/p + 1/q = 1$. Then if $l$ is a bounded linear functional on $L^p(E, d\mu)$ where $\mu$ is a $\sigma$-finite measure, $l(f) = \int_Efgd\mu$ for some $g \in L^q(E,d\mu)$...
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Is the map sends $T$ to $T^*$ adjoint of $T$ surjective?
Let $B(X)$ denotes the set of all bounded linear operators from $X$ to $X$, where $X$ is a Banach space. Same is defined for the set $B(X^*)$, where $X^*$ denotes the set of all bounded linear ...
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Showing $X^*$ is separable implies $X$ is separable using the Riesz lemma
If $X$ is a Banach space and $X^*$ is separable, then $X$ is
separable.
Here, David Mitra mentions a proof using the Riesz lemma. However, I do not fully understand it.
You could also use Riesz' ...
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Double dual space is isomorphic to vector space - Intuition
The recent topics I studied were linear functionals and dual spaces. I like to think about a linear functional as a stack of hyperplanes like it is described here.
In "Finite dimensional vector ...
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Further interesting examples? Obtaining (co)monoids from dual objects
1. Context
Obtaining (co)monoids from dual objects
Let $(C, \otimes, I, a, l,r)$ be a monoidal category. To simplify notation (and work with string diagrams) we assume that $C$ is strict. Let $V \in ...
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Does existence of some (nice) non-trivial functionals in $\ell_\infty^*\setminus\ell_1$ give a free ultrafilter on $\omega$?
If we there is a free ultrafilter $\mathscr U$ on $\omega$ then
$$\newcommand{\Ulim}{\operatorname{{\mathscr U}-lim}}f \colon x = (x_n) \mapsto \Ulim x_n$$
defines a functional belonging to $\ell_\...
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Prove that $W$ is $T$-invariant if and only if $W^0$ is $T^t$-invariant.
Let $T$ be a linear operator on a finite-dimensional vector space $V$, and let $W$ be a subspace of $V$. Let $W^0 \subset V^*$ be the annihilator of $W$. Prove that $W$ is $T$-invariant if and only if ...
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Intuition of dual space action in dual of $\ell_p$
For $p>1,$ the sequence space $\ell_p$ is defined as
$$\ell_p=\left\{ x=(x_i)_{i\in\mathbb{N}} : \sum_{i=1}^\infty |x_i|^p < \infty \right\}.$$
A classical duality theorem of $\ell_p$ space is ...
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Motivation for the development of the double dual of a vector space [duplicate]
I was reading on the double dual of a vector space $V$ recently. I was wondering what applications (within mathematics) there are for this concept and/or what was the motivation for the development of ...
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