Questions tagged [dual-maps]
54 questions
6
votes
1 answer
Can someone explain why we define adjoint?
Recently I've been reviewing linear algebra. The definition of adjoint of a linear map on an inner product space seems not really natrual. Looks like people use this to define normal and self-adjoint and prove the spectral theorem. But from the…
jiefu hou
- 71
4
votes
2 answers
Spectrum of an operator is approximate point spectrum plus spectrum of dual operator
I'm trying to show that given an operator $T \in B(X)$ with $X$ Banach we have $$\sigma(T) = \sigma_{ap}(T) \cup \sigma_p(T') $$
Where $T' \in B(X')$ is the dual operator.
I know that $\sigma(T) = \sigma(T')$ and certainly $\sigma_{ap}(T) \subset…
Wooster
- 3,885
3
votes
1 answer
Prove that if $\phi$ is injective then the dual function $\phi^{*}$ is surjective
V,W finite dimensional K-vector spaces.
$\phi:V \rightarrow W$ linear map
I have proven that if $\phi$ is surjective then $\phi^{*}$ is injective. Now I have to prove that if $\phi$ is injective then $\phi^{*}$ is surjective and I don't have a clue…
user900336
3
votes
4 answers
The transpose of a linear injection is surjective.
Let
$$T:V\longrightarrow W$$
be a linear map (of vector spaces), and let
\begin{eqnarray}
T^*:W^* &\longrightarrow& V^* \\
f\ &\longmapsto& f\circ T
\end{eqnarray}
be its transpose (or dual map).
How to show that if $T$ is injective then $T^*$…
Spenser
- 20,135
3
votes
0 answers
My attempts to show the dual map is isometric.
Theorem: Let $X$ and $Y$ be normed spaces such that $X\cong Y$. Let $\phi:X\rightarrow Y$ be an isometric isomorphism. Then the dual map ${\phi}^*:{Y}^*\rightarrow{X}^{*},\lambda\mapsto\lambda\circ\phi$ is an isometric isomorphism.
I have already…
Answer Lee
- 1,069
3
votes
1 answer
Prove that $\Omega(x^{*^1}, \ldots, x^{*^n}; x_1, \ldots, x_n) = \phi (x^{*^1}, \ldots, x^{*^n}) \Delta(x_1, \ldots, x_n)$
In the book Linear Algebra by Werner Greub, at page $112$, it is given that,
Let $E^*, E$ be a pair of dual vector spaces and $\Delta^* \not = 0,
\Delta \not = 0$ be determinant functions in $E^*$ and $E$. It will be
shown that …
Our
- 7,425
2
votes
2 answers
About the range of an operator and its adjoint
I need a hand with the proof of this result:
If we have an operator between Banach spaces $T:X\to Y$, with closed range, then the adjoint operator $T^*:Y^*\to X^*$ has also closed range.
Thanks in advance for any help.
Mark_Hoffman
- 1,579
2
votes
1 answer
Relation between the elementary Rouché-Fröbenius theorem and the more abstract Fröbenius theorem with exact sequences.
In highschool I was taught that the "Rouché-Fröbenius theorem" was that given a homogenous linear system of equations, the solution space has dimension $n-r$ where $n$ is the dimension of total space (number of variables) and $r$ is the number of…
2
votes
0 answers
What’s the intuition for these annihilator results in linear algebra?
I’m studying dual spaces in linear algebra.
I have proved the following two results. Note: I say $U^0$ for the annihilation of $U$.
For subspaces $U,W$ of a vector space $V$,
$(U+W)^0=U^0\cap W^0$
And
$U^0+W^0\subseteq (U\cap W)^0$
With the…
jet
- 507
2
votes
1 answer
The property of duality map in case the dual space is strictly convex
Let $(E, | \cdot |)$ be a Banach space and $(E', \| \cdot \|)$ its dual. Assume that $E'$ is strictly convex. Then for each $x\in E$, there is a unique $f_x \in E'$ such that $\|f_x\|=|x|$ and $\langle f_x, x\rangle = |x|^2$. Clearly, $f_{\lambda x}…
Analyst
- 6,351
2
votes
1 answer
Is there standard mathematical terminology for structures that come in one-to-one pairs?
Background / concrete example of what I'm asking about:
Given a set $X$ and a set of "open" sets $\mathcal T\subseteq\mathcal P(X)$ satisfying the open-set axioms (of topological spaces), we can immediately define a closure operator…
WillG
- 7,382
2
votes
1 answer
Derivation of Dual Curve for Parametric Equations
I am studying projective geometry and am stuck on understanding the following:
For a parametric curve
$$
x = x(t), y = y(t)
$$
the dual curve is given by
$$
X=\frac{-y′}{xy′-yx'},
Y=\frac{x′}{xy′−yx′}
$$
This is explained on Wikipedia, in Chapter 1,…
adam.hendry
- 352
2
votes
1 answer
Is that true that for every linear transformation $\phi : V^* \to W^*$ there is a linear transformation $\psi: W \to V$ such that $\psi^* = \phi$?
$V$ and W are finitely dimensional linear spaces over the field $K$. Is that true that for every linear transformation $\phi : V^* \to W^*$ there is a linear transformation $\psi: W \to V$ such that $\psi^* = \phi$?
W* means dual space of W, that…
mymathc
- 191
2
votes
1 answer
Is it true that $\dim({\rm im}(f))=\dim({\rm im}(f^{*}))$?
I have a question regarding dimensions of finite vectors spaces.
Let $f$ be a linear map between two vector spaces $f:E_{1}\longrightarrow E_{2}$ with dimensions $m$ and $n$ respectively in a field $\mathbb{F}$ and let $E_{1}^{*}$ and $E_{2}^{*}$ be…
Tutusaus
- 667
2
votes
2 answers
Proving that $\phi (T) = T^*$ is an isomorphism between vector spaces
I am tasked with the following:
I am thus tasked with proving:
$1)$ $\phi(T)$ is linear, so that it respects closure under scalar multiplication and addition.
$2)$ $\phi(T)$ is a bijection.
I only need justification as to whether or not my…
sangstar
- 1,995
- 2
- 21
- 44