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 $k$.

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 linear algebra. I was wondering if anyone knew a…
WWright
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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 vector space $V=\mathbb F^{(\kappa)}$ (that is an…
Asaf Karagila
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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 double duals; they are isomorphic to their duals as…
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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 functionals.
omar
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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 and theories.
Theorem
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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 existence of lambda notation. More specifically,…
Noa
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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 $y\in\ell^1$. We claim that $f_y(x)=\sum_{k=1}^\infty…
Laars Helenius
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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}^{\infty} x_ky_k$ is not a linear bounded functional on…
Benzio
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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 supposed to use the following hint: First show that for…
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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: For every $\epsilon >0$ I wanted to show the…
user24367
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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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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 in a Banach space $E$ to the double continuous…
sjvega
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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 every vector space V we make correspond the dual…
Comic Book Guy
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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 a Banach space. Is there a good book where I can…
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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: $$f_1(x,y,z) = x - 2y;\quad f_2(x,y,z) = x + y + z; \quad f_3(x,y,z) = y - 3z.$$ Prove that $\{f_1, f_2,…
Math Student
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