Question

Show that the dual space of the vector space of all polynomials is isomorphic to the infinite-dimensional Euclidean vector space over the reals

score 2.9k · Answer 1 · 2016-09-17 16:39:50

2.9k

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Show that the dual space of the vector space of all polynomials is isomorphic to the infinite-dimensional Euclidean vector space over the reals

Published on 17 Sep 2016 - 16:39

score 61 · Answer 2 · 2016-09-22 16:43:21

61

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Let $T:P_n(\mathbb R) \rightarrow \mathbb R^{n+1}$ be defined by $T(f)=(f(x_1), f(x_2). ..., f(x_n), f(x_{n+1}))$. Prove that T is an isomorphism.

Published on 22 Sep 2016 - 16:43

score 470 · Answer 3 · 2016-09-26 22:02:51

470

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A bijection between two different basis for different vector spaces

Published on 26 Sep 2016 - 22:02

score 189 · Answer 4 · 2016-10-01 22:02:46

189

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Where did I use the assumption of finite-dimensional vector space? Isomorphism in a quotient space

Published on 01 Oct 2016 - 22:02

score 327 · Answer 5 · 2016-10-03 18:15:37

327

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What are the implications of the fact that vector spaces are isomorphic?

Published on 03 Oct 2016 - 18:15

score 219 · Answer 6 · 2016-10-24 23:49:22

219

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A norm generated by the quadratic form of T implies T is an isomorphism.

Published on 24 Oct 2016 - 23:49

score 207 · Answer 7 · 2016-11-05 00:29:28

207

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Prove that $(\Bbb{R}^n,\| \cdot \|_2)$ and $(\Bbb{R}^n,\| \cdot \|_\infty)$ are not isometrically isomorphic, if $n\geq 2$

Published on 05 Nov 2016 - 0:29

score 896 · Answer 8 · 2026-02-23 06:42:05

896

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How to calculate angle between two points in 3d space when the three axis changes

Published on 23 Feb 2026 - 6:42

score 1k · Answer 9 · 2016-11-23 09:27:33

1k

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An example of a Banach space isomorphic but not isometric to a dual Banach space

Published on 23 Nov 2016 - 9:27

score 113 · Answer 10 · 2016-12-03 17:32:59

113

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Show that $L$ is an isomorhism iff $L\circ L$ is an isomorphism too.

Published on 03 Dec 2016 - 17:32

score 184 · Answer 11 · 2016-12-04 00:53:01

184

Views

Show there exists an isomorphism

Published on 04 Dec 2016 - 0:53

score 520 · Answer 12 · 2016-12-04 13:25:03

520

Views

Does every finite dimensional real vector space have a canonical basis?

Published on 04 Dec 2016 - 13:25

score 595 · Answer 13 · 2016-12-17 02:38:00

595

Views

Why is it okay to think of naturally isomorphic sets as being equal to each-other?

Published on 17 Dec 2016 - 2:38

score 246 · Answer 14 · 2016-12-25 18:33:44

246

Views

Sobolev spaces embeddings & isomorphisms

Published on 25 Dec 2016 - 18:33

score 824 · Answer 15 · 2016-12-26 04:42:50

824

Views

What does it mean that interior of a simplex is a vector space?

Published on 26 Dec 2016 - 4:42

List Question

Show that the dual space of the vector space of all polynomials is isomorphic to the infinite-dimensional Euclidean vector space over the reals

Let $T:P_n(\mathbb R) \rightarrow \mathbb R^{n+1}$ be defined by $T(f)=(f(x_1), f(x_2). ..., f(x_n), f(x_{n+1}))$. Prove that T is an isomorphism.

A bijection between two different basis for different vector spaces

Where did I use the assumption of finite-dimensional vector space? Isomorphism in a quotient space

What are the implications of the fact that vector spaces are isomorphic?

A norm generated by the quadratic form of T implies T is an isomorphism.

Prove that $(\Bbb{R}^n,\| \cdot \|_2)$ and $(\Bbb{R}^n,\| \cdot \|_\infty)$ are not isometrically isomorphic, if $n\geq 2$

How to calculate angle between two points in 3d space when the three axis changes

An example of a Banach space isomorphic but not isometric to a dual Banach space

Show that $L$ is an isomorhism iff $L\circ L$ is an isomorphism too.

Show there exists an isomorphism

Does every finite dimensional real vector space have a canonical basis?

Why is it okay to think of naturally isomorphic sets as being equal to each-other?

Sobolev spaces embeddings & isomorphisms

What does it mean that interior of a simplex is a vector space?

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