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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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
Kenneth.K
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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 a subset of $V$, the annihilator of $S$ is the set…
user279515
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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 between $X$ and $Y^*$)?
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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 wondering whether on the level of group and group theory…
MarcusH
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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 $n$ respectively. (Here $p$, $q$ are conjugate indices…
Jack
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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 $K$-vector spaces belonging to $\mathcal U$, and let $i\ge0$ be an integer.…
Pierre-Yves Gaillard
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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 Banach algebra with the norm $$||f|| :=…
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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 space to a tangent space. What I'd like to know is…
Rajesh D
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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 between $y_i = \phi(e_i)\in \ell_{q}$ and $\phi$…
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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 well defined, linear, injective, and continuous in…
Bob
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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 and the idea of proof.
Vladislav
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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 $$\|Y\|_* = \sup \left\{ \mathbb{E}[X \cdot Y] ~:~ \forall t \in \mathbb{R} ~~ \mathbb{E}[e^{tX}] \le e^{t^2/2}\right\},$$ where the supremum is…
Thomas
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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 isomorphic to $\ell_{\infty}$ as in class we haven't gone…
Matt
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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 each $F \in X''$, $$\sum_{n = 1}^\infty | F(f_n) | \le…
namasikanam
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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 immersion) of $X$ into $X^{**}. $ Lemma 1: Let $X$ be a…
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