Question

Given $T \in L(X,Y)$, show the equivalence between: existence of $S$ such that $S(T(x))=x$, and $T$ being injective with $T(X)$ complemented in $Y$

score 678 · Answer 1 · 2019-01-07 15:39:02

678

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Given $T \in L(X,Y)$, show the equivalence between: existence of $S$ such that $S(T(x))=x$, and $T$ being injective with $T(X)$ complemented in $Y$

Published on 07 Jan 2019 - 15:39

score 962 · Answer 2 · 2026-03-25 19:01:18

962

Views

Open Mapping Theorem fails when space is not complete

Published on 25 Mar 2026 - 19:01

score 107 · Answer 3 · 2019-01-09 16:34:22

107

Views

Prove that (, ) is a complete metric space

Published on 09 Jan 2019 - 16:34

score 52 · Answer 4 · 2019-01-17 10:02:00

52

Views

Rudin's functional analysis appendix A4 (a)

Published on 17 Jan 2019 - 10:02

score 45 · Answer 5 · 2019-01-18 11:51:50

45

Views

Proof completenes of $ \{x \in \mathbb{C}^\mathbb{N}\ |\ \sum_{n=1}^\infty s_n |x_n|^p < \infty \}$

Published on 18 Jan 2019 - 11:51

score 123 · Answer 6 · 2019-01-20 11:49:04

123

Views

Subsets of complex plane with euclidean metric that are complete

Published on 20 Jan 2019 - 11:49

score 355 · Answer 7 · 2019-01-23 20:14:53

355

Views

Completion of inner product space

Published on 23 Jan 2019 - 20:14

score 1.3k · Answer 8 · 2019-01-27 06:01:36

1.3k

Views

Critique my proof: Proving that $C[a,b]$ is not a Hilbert Space under $L^2$ norm.

Published on 27 Jan 2019 - 6:01

score 2.7k · Answer 9 · 2026-02-23 12:07:13

2.7k

Views

Show that the class $C_c(\mathbb{R^n})$ of continuous functions with compact support is not a complete metric space

Published on 23 Feb 2026 - 12:07

score 3.4k · Answer 10 · 2019-02-06 11:06:37

3.4k

Views

$c_{00}$ is not complete

Published on 06 Feb 2019 - 11:06

score 699 · Answer 11 · 2019-02-12 06:34:16

699

Views

completeness definition (real analysis basic) (many edits)

Published on 12 Feb 2019 - 6:34

score 1.7k · Answer 12 · 2019-02-16 23:33:37

1.7k

Views

Normed space $C^2[0,1]$ with norm $\lVert f\rVert:=\max_{t\in[0,1]}\{\lvert f(t)\rvert+\lvert f''(t)\rvert\}$ is Banach space

Published on 16 Feb 2019 - 23:33

score 253 · Answer 13 · 2026-03-25 01:27:58

253

Views

Completeness of the tangent bundle of riemannian manifold

Published on 25 Mar 2026 - 1:27

score 65 · Answer 14 · 2019-02-18 16:19:04

65

Views

Does there exist an intermediate step of completion of the algebraic closure $\bar{\mathbb{R}}$ to get the field $\mathbb{C}$?

Published on 18 Feb 2019 - 16:19

score 1.4k · Answer 15 · 2019-02-19 20:26:05

1.4k

Views

The space of all finite-degree polynomials $\mathbb{P}$ is not complete in any norm.

Published on 19 Feb 2019 - 20:26

List Question

Given $T \in L(X,Y)$, show the equivalence between: existence of $S$ such that $S(T(x))=x$, and $T$ being injective with $T(X)$ complemented in $Y$

Open Mapping Theorem fails when space is not complete

Prove that (, ) is a complete metric space

Rudin's functional analysis appendix A4 (a)

Proof completenes of $ \{x \in \mathbb{C}^\mathbb{N}\ |\ \sum_{n=1}^\infty s_n |x_n|^p < \infty \}$

Subsets of complex plane with euclidean metric that are complete

Completion of inner product space

Critique my proof: Proving that $C[a,b]$ is not a Hilbert Space under $L^2$ norm.

Show that the class $C_c(\mathbb{R^n})$ of continuous functions with compact support is not a complete metric space

$c_{00}$ is not complete

completeness definition (real analysis basic) (many edits)

Normed space $C^2[0,1]$ with norm $\lVert f\rVert:=\max_{t\in[0,1]}\{\lvert f(t)\rvert+\lvert f''(t)\rvert\}$ is Banach space

Completeness of the tangent bundle of riemannian manifold

Does there exist an intermediate step of completion of the algebraic closure $\bar{\mathbb{R}}$ to get the field $\mathbb{C}$?

The space of all finite-degree polynomials $\mathbb{P}$ is not complete in any norm.

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