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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Matrix representation of adjoint & co-adjoint orbit of $so(3)$

So I am trying to find the co-adjoint orbits of the lie algebra $so(3)^*$ from this example but I am stuck with a very trivial linear algebra property now I found the adjoint orbits and I know the matrix representation of the adjoint action/map is…
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Is $\dim \mathcal{L}(V,W)\ge |F|$ for infinite dimensional vector spaces $V$ and $W$?

In Lemma 2 of this answer, it was shown that if $\dim V$ was infinite, then $\dim V^*\ge |F|$. As $V^*=\mathcal{L}(V,F)$, this led me to wonder if the same held for the general space of linear transformations $\mathcal{L}(V,W)$. More concretely, if…
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proof of surjectivity of dualT implies injectivity of T and conversely

This is from Axler's Linear Algebra book page 107 and 108. If $T$ is a linear map from $V$ to $W$, and $T^*$ is a dual map from $W^*$ to $V^*$, Why is range $T$ = $W$ (surjectivity of $T$) a necessary and sufficient condition for (range $T$)$^0 =…
David Kwak
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What is the dual space of $C[0,1]$?

I want to know more about $BV[0,1]$. Like the way a function in $BV[0,1]$ acts on $C[0,1]$, and when a sequence in $C[0,1]$ is weakly convergence?
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Let $w$ be a positive continuous function for which $\int_0^1 w(x)dx = \int_0^1 x^2w(x)dx = 1$. Prove that $\int_0^1 xw(x)dx < 1$.

Let $w$ be a positive continuous function for which $$\int_0^1 w(x)dx = \int_0^1 x^2w(x)dx = 1.$$ Prove that $\int_0^1 xw(x)dx < 1$. I was thinking of using the Reisz Representation Theorem for this but my intuitive understanding of it is lacking. I…
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Finite dimensional dual space

Let $X$ be a normed linear space with a finite dimensional dual $X^*$. How do I prove X is also finite dimensional?
nonpara
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Why is $L^q$ congruent to $(L^p)^*$?

Why is $L^q$ congruent to $(L^p)^*$ for $1\leq p < \infty$? Here $q=\frac{p}{p-1}$.
Raphaël
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write function as sum of covectors

I've got what should be a simple question. I have a function f(x,y) = 3x + 2y The question asks to write this function as a sum of dual vectors. Any help on where to begin?
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Annihilator in dual space

Let U and W are subspaces of a vector space V. If U is subset of W, inh(W) is subset of inh(U). Is the Converse true? How?
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show that there exists a basis $ \ \mathcal{B}'=\{e_1',e_2',......, e_n' \} \ \ of \ \ V $ such that $ \ G(e_i,e_j')=\delta^i_j \ $

Let $ V \ $ be a n-dimensional vector space over the field $ \ \mathbb{F} \ $ and let $ \ G \ $ be a non-degenerate billinear form on $ \ V \ $. Thus the map $ \ L_G : V \to V^{*} \ $ is a linear Isomorphism. (a) Let $ \…
MAS
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What kinds of structure are there of the elements in the double dual space $V^{**}$?

Let $V=\mathbb{R}^3$ be a vector space over $\mathbb{R}$. The dual space of $V$ contains the elements of the form $ax+by+cz$. What kinds of structure are there of the elements in the double dual space $V^{**}$?
Infinite
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Unbounded Operator not bounded on any ball

I came across a small proof where the following implication was used: Let $X$ be a normed v.s. and $T \in X'$ an unbounded operator ( $X'$ denotes the dual of $X$ ), i.e. $$ \| T (x) \| \gt M \| x \| \qquad \forall M >0, x \in X $$ then $$…
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Identification of $\ell_1^n$ $(\ell_\infty^n)$ with $\ell_\infty^{n^*}$ $(\ell_1^{n^*})$.

Let $X=\ell_1^n$ or $\ell_\infty^n$. Then any member of $\ell_1^n$ ($\ell_\infty^n$) can be identified as a functional over $\ell_\infty^n$ ($\ell_1^n$) via the canonical isometric isomorphism $\psi$. What does this identification look like? For…
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Prove A=A* where $Ax = \sum_{n}a_{n}\langle x, u_n \rangle u_n =0 $ iff $a_{n} \in \mathbb{R}$

How to prove A=A* where $Ax = \sum_{n}a_{n}\langle x, u_n \rangle u_n=0 $ iff $a_{n} \in \mathbb{R}$ and $[u_n]$ is an orthonormal sequence? Edit: does it have something to do with the equality: $\langle Ax, y \rangle = \langle x, A^*y \rangle =…
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Why is the topology of the norm of dual of Banach space $X$ identical to that of uniform convergence on the bounded subset of $X$?

Let $X$ be a Banach space; the associated dual space is denoted by $X^∗$. Why is the topology of the norm of $X^*$ identical to that of uniform convergence on the bounded subset of $X$? An idea please.
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