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 681 · Answer 1 · 2026-04-15 20:39:04

681

Views

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 15 Apr 2026 - 20:39

score 963 · Answer 2 · 2026-05-10 15:44:36

963

Views

Open Mapping Theorem fails when space is not complete

Published on 10 May 2026 - 15:44

score 110 · Answer 3 · 2026-04-14 02:28:54

110

Views

Prove that (, ) is a complete metric space

Published on 14 Apr 2026 - 2:28

score 55 · Answer 4 · 2026-04-10 13:02:43

55

Views

Rudin's functional analysis appendix A4 (a)

Published on 10 Apr 2026 - 13:02

score 48 · Answer 5 · 2026-04-14 13:31:37

48

Views

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

Published on 14 Apr 2026 - 13:31

score 126 · Answer 6 · 2026-04-15 06:28:28

126

Views

Subsets of complex plane with euclidean metric that are complete

Published on 15 Apr 2026 - 6:28

score 359 · Answer 7 · 2026-05-10 13:28:56

359

Views

Completion of inner product space

Published on 10 May 2026 - 13:28

score 1.3k · Answer 8 · 2026-04-11 23:45:54

1.3k

Views

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

Published on 11 Apr 2026 - 23:45

score 2.7k · Answer 9 · 2026-04-12 23:08:58

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 12 Apr 2026 - 23:08

score 3.4k · Answer 10 · 2026-04-15 05:08:47

3.4k

Views

$c_{00}$ is not complete

Published on 15 Apr 2026 - 5:08

score 702 · Answer 11 · 2026-04-13 18:31:54

702

Views

completeness definition (real analysis basic) (many edits)

Published on 13 Apr 2026 - 18:31

score 1.7k · Answer 12 · 2026-04-16 09:39:00

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 Apr 2026 - 9:39

score 256 · Answer 13 · 2026-05-11 00:32:42

256

Views

Completeness of the tangent bundle of riemannian manifold

Published on 11 May 2026 - 0:32

score 68 · Answer 14 · 2026-04-14 06:17:19

68

Views

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

Published on 14 Apr 2026 - 6:17

score 1.4k · Answer 15 · 2026-04-13 13:00:01

1.4k

Views

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

Published on 13 Apr 2026 - 13:00

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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