Question

If W is a subspace of V and $x \notin W$, prove that there exists $f \in W^0$ such that $f(x) \neq 0$.

score 805 · Answer 1 · 2020-04-05 17:08:38

805

Views

If W is a subspace of V and $x \notin W$, prove that there exists $f \in W^0$ such that $f(x) \neq 0$.

Published on 05 Apr 2020 - 17:08

score 270 · Answer 2 · 2020-04-05 19:07:08

270

Views

Bound on the norm of a bounded linear functional $f:C[0,1] \rightarrow \mathbb{R}$ defined by $f(\varphi)=\int_0^1\varphi(x)dx$.

Published on 05 Apr 2020 - 19:07

score 833 · Answer 3 · 2020-04-06 04:22:46

833

Views

How to prove that $(S^{0})^{0} = \operatorname{span}(\psi(S))$, where $\psi: V \to V^{**}$ is the natural isomorphism

Published on 06 Apr 2020 - 4:22

score 96 · Answer 4 · 2026-03-22 06:34:21

96

Views

Prove that $\exists f\in V^*$ and $w\in W$ s.t. $A(v)=f(v)w\;\forall v\in V$

Published on 22 Mar 2026 - 6:34

score 354 · Answer 5 · 2026-03-26 03:11:36

354

Views

Why is it standard convention to denote dual vectors as row vectors when using coordinates?

Published on 26 Mar 2026 - 3:11

score 49 · Answer 6 · 2020-04-07 00:54:56

49

Views

About dual space and linear functional

Published on 07 Apr 2020 - 0:54

score 47 · Answer 7 · 2020-04-07 01:15:58

47

Views

About a theorem of dual space

Published on 07 Apr 2020 - 1:15

score 182 · Answer 8 · 2020-04-07 03:26:52

182

Views

deduce that there exists a unique polynomial q(x) of degree at most n such that$ q(c_i)=a_i$ for $0 \leq i \leq n$.

Published on 07 Apr 2020 - 3:26

score 1.1k · Answer 9 · 2020-04-07 03:58:26

1.1k

Views

prove that if W is a subspace of V, then dim(W)+dim(W^0)=dim(V)

Published on 07 Apr 2020 - 3:58

score 562 · Answer 10 · 2020-04-08 01:02:56

Let $V=R^3$, and define ${f_1,f_2,f_3} \in V^*$ as follows: $f_1(x,y,z)=x-2y, f_2(x,y,z)=x+y+z$, prove that ${f_1,f_2,f_3}$ is a basis for $V^*$.

score 66 · Answer 11 · 2020-04-08 03:50:40

Show that the plane $\{su+tv\mid s,t \in\Bbb R\}$ through the origin in $\Bbb R^3$ is equal to the null space of some element of $(\Bbb R^3)^{*}$.

score 67 · Answer 12 · 2020-04-08 05:26:44

67

Views

show that there exist unique polynomials $p_0(x),...,p_n(x)$ such that $p_i(c_j)=\delta_{ij}$ for $0 \leq i,j \leq n$.

Published on 08 Apr 2020 - 5:26

score 195 · Answer 13 · 2026-03-25 19:04:49

195

Views

Dual space of the Intersection of locally convex vector spaces

Published on 25 Mar 2026 - 19:04

score 25 · Answer 14 · 2020-04-09 14:49:52

Why is the topology of the norm of dual of Banach space $X$ identical to that of uniform convergence on the bounded subset of $X$?

score 209 · Answer 15 · 2020-04-09 23:15:22

209

Views

let $T:V \rightarrow W$ be linear map. Prove that $T^t$ is onto if and only if T is one-to-one.

Published on 09 Apr 2020 - 23:15

List Question

If W is a subspace of V and $x \notin W$, prove that there exists $f \in W^0$ such that $f(x) \neq 0$.

Bound on the norm of a bounded linear functional $f:C[0,1] \rightarrow \mathbb{R}$ defined by $f(\varphi)=\int_0^1\varphi(x)dx$.

How to prove that $(S^{0})^{0} = \operatorname{span}(\psi(S))$, where $\psi: V \to V^{**}$ is the natural isomorphism

Prove that $\exists f\in V^*$ and $w\in W$ s.t. $A(v)=f(v)w\;\forall v\in V$

Why is it standard convention to denote dual vectors as row vectors when using coordinates?

About dual space and linear functional

About a theorem of dual space

deduce that there exists a unique polynomial q(x) of degree at most n such that$ q(c_i)=a_i$ for $0 \leq i \leq n$.

prove that if W is a subspace of V, then dim(W)+dim(W^0)=dim(V)

Let $V=R^3$, and define ${f_1,f_2,f_3} \in V^$ as follows: $f_1(x,y,z)=x-2y, f_2(x,y,z)=x+y+z$, prove that ${f_1,f_2,f_3}$ is a basis for $V^$.

Show that the plane $\{su+tv\mid s,t \in\Bbb R\}$ through the origin in $\Bbb R^3$ is equal to the null space of some element of $(\Bbb R^3)^{*}$.

show that there exist unique polynomials $p_0(x),...,p_n(x)$ such that $p_i(c_j)=\delta_{ij}$ for $0 \leq i,j \leq n$.

Dual space of the Intersection of locally convex vector spaces

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 $T:V \rightarrow W$ be linear map. Prove that $T^t$ is onto if and only if T is one-to-one.

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