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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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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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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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 ...