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 $k$.
Questions tagged [dual-spaces]
1457 questions
264
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7 answers
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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119
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3 answers
Why are infinitely dimensional 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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114
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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…
Ben Blum-Smith
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103
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2 answers
Can we identify the duals of $\ell^\infty$ and $L^{\infty}$ with another "natural space"?
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
- 2,651
65
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3 answers
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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54
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2 answers
When exactly is the dual of $L^1$ isomorphic to $L^\infty$ via the natural map?
The dual space to the Banach space $L^1(\mu)$ for a sigma-finite measure $\mu$ is $L^\infty(\mu)$, given by the correspondence
$\phi \in L^\infty(\mu) \mapsto I_\phi$, where $I_\phi(f) = \int f \cdot \phi \,d\mu$ for $f \in L^1(\mu)$.
Now, we…
Ben Passer
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2 answers
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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49
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4 answers
A Banach space is reflexive if its dual is reflexive (and conversely)
How to show that a Banach space $X$ is reflexive if its dual $X'$ is reflexive?
Ken
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46
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0 answers
Regarding metrizability of weak/weak* topology and separability of Banach spaces.
Let $X$ be a Banach space, $X^*$ its dual, $\mathcal{B}$ the unit ball of $X$ and $\mathcal{B}^*$ the unit ball of $X^*$. The following result is well-known
Theorem. If $X$ is separable, then $\mathcal{B}^*$ endowed with the weak*-topology is…
user160185
41
votes
1 answer
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 $\: \mathbf{B}_0 = \mathbf{B} \:$. For all non-negative integers…
user57159
37
votes
3 answers
Let $c\subset\ell^\infty$ be the subspace of sequences that converge. How to prove that 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
- 2,267
26
votes
1 answer
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 of $c$ whose dual is isometrically isomorphic to…
Nirakar Neo
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25
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2 answers
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…
Rudy the Reindeer
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