Question

Is it true that $\dim({\rm im}(f))=\dim({\rm im}(f^{*}))$?

score 187 · Answer 1 · 2026-03-21 14:25:32

187

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Is it true that $\dim({\rm im}(f))=\dim({\rm im}(f^{*}))$?

Published on 21 Mar 2026 - 14:25

score 374 · Answer 2 · 2026-03-24 12:15:10

374

Views

Proving $\operatorname{coker}(f^) \cong (\ker f)^$ for a linear map $f$

Published on 24 Mar 2026 - 12:15

score 83 · Answer 3 · 2020-05-31 18:20:11

83

Views

What is the dual of the space $V=\{f: K\to \mathbb R^n, f(x)=Ax+b\}$ with $K\subset\mathbb R^n$ compact?

Published on 31 May 2020 - 18:20

score 108 · Answer 4 · 2020-06-16 09:01:37

108

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Assume that $ \pi $ is a projection operator in $ E $. Prove that $ \pi^{} $ is a projection operator in $ E^{} $.

Published on 16 Jun 2020 - 9:01

score 39 · Answer 5 · 2020-07-07 19:14:08

39

Views

Is the equivalent function between isomorphic domains called a dual?

Published on 07 Jul 2020 - 19:14

score 39 · Answer 6 · 2026-03-10 02:18:37

39

Views

If two elements are different there is a functional under where the image is different

Published on 10 Mar 2026 - 2:18

score 83 · Answer 7 · 2023-05-05 20:19:21

83

Views

Adjoint of the Dual Map

Published on 05 May 2023 - 20:19

score 92 · Answer 8 · 2023-05-06 10:58:35

92

Views

What’s the intuition for these annihilator results in linear algebra?

Published on 06 May 2023 - 10:58

score 34 · Answer 9 · 2023-08-08 13:30:44

34

Views

How do we know $T'(\psi_j)=\psi_j\circ T$, where $T:V\to W$, $T': W'\to W'$ is dual map of $T$, and $\psi_j\in W'$?

Published on 08 Aug 2023 - 13:30

score 126 · Answer 10 · 2024-02-09 14:14:46

126

Views

$\text{span}(\phi_1, \cdots \phi_m) = \{\bigcap_{i=1}^m \text{null}(\phi_i)\}^0$

Published on 09 Feb 2024 - 14:14

score 20 · Answer 11 · 2024-03-08 09:05:04

20

Views

Dual map-injectivity implies surjection

Published on 08 Mar 2024 - 9:05

score 22 · Answer 12 · 2024-03-10 20:00:42

22

Views

Dual mappings - from U to V and from V to W

Published on 10 Mar 2024 - 20:00

score 79 · Answer 13 · 2024-03-21 01:40:48

79

Views

Prove $T' = 0$ if and only if $T=0$

Published on 21 Mar 2024 - 1:40

score 883 · Answer 14 · 2015-04-30 15:40:33

883

Views

Spectrum of an operator is approximate point spectrum plus spectrum of dual operator

Published on 30 Apr 2015 - 15:40

score 87 · Answer 15 · 2017-08-04 17:04:57

87

Views

Prove that $\Omega(x^{^1}, \ldots, x^{^n}; x_1, \ldots, x_n) = \phi (x^{^1}, \ldots, x^{^n}) \Delta(x_1, \ldots, x_n)$

Published on 04 Aug 2017 - 17:04

List Question

Is it true that $\dim({\rm im}(f))=\dim({\rm im}(f^{*}))$?

Proving $\operatorname{coker}(f^) \cong (\ker f)^$ for a linear map $f$

What is the dual of the space $V=\{f: K\to \mathbb R^n, f(x)=Ax+b\}$ with $K\subset\mathbb R^n$ compact?

Assume that $ \pi $ is a projection operator in $ E $. Prove that $ \pi^{} $ is a projection operator in $ E^{} $.

Is the equivalent function between isomorphic domains called a dual?

If two elements are different there is a functional under where the image is different

Adjoint of the Dual Map

What’s the intuition for these annihilator results in linear algebra?

How do we know $T'(\psi_j)=\psi_j\circ T$, where $T:V\to W$, $T': W'\to W'$ is dual map of $T$, and $\psi_j\in W'$?

$\text{span}(\phi_1, \cdots \phi_m) = \{\bigcap_{i=1}^m \text{null}(\phi_i)\}^0$

Dual map-injectivity implies surjection

Dual mappings - from U to V and from V to W

Prove $T' = 0$ if and only if $T=0$

Spectrum of an operator is approximate point spectrum plus spectrum of dual operator

Prove that $\Omega(x^{^1}, \ldots, x^{^n}; x_1, \ldots, x_n) = \phi (x^{^1}, \ldots, x^{^n}) \Delta(x_1, \ldots, x_n)$

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