Question

Prove that the set $\{x\in \Bbb R^n:\|x\|_1=1\}$ is compact .

score 1k · Answer 1 · 2026-04-16 20:52:34

1k

Views

Prove that the set $\{x\in \Bbb R^n:\|x\|_1=1\}$ is compact .

Published on 16 Apr 2026 - 20:52

score 903 · Answer 2 · 2026-04-16 11:06:26

903

Views

Prove that the set of all unit vectors in a finite-dimensional normed vector space is compact.

Published on 16 Apr 2026 - 11:06

score 71 · Answer 3 · 2026-04-14 19:11:59

71

Views

If $A$ is a continuous linear operator on $l^2$ such that $||A − T|| < 1$, show that $A$ is not invertible.

Published on 14 Apr 2026 - 19:11

score 134 · Answer 4 · 2026-04-11 14:38:34

134

Views

A question about uniformly bounded set of operators

Published on 11 Apr 2026 - 14:38

score 350 · Answer 5 · 2026-04-09 20:19:30

350

Views

Hyperplane separation theorem

Published on 09 Apr 2026 - 20:19

score 158 · Answer 6 · 2026-04-15 22:40:53

158

Views

Prove that the set of functions $\sin kx$ is closed and bounded

Published on 15 Apr 2026 - 22:40

score 866 · Answer 7 · 2026-04-15 18:17:34

866

Views

Counter example to show convergence in $l^\infty$ metric does not imply the same in $l^1$ metric on a specified metric space

Published on 15 Apr 2026 - 18:17

score 648 · Answer 8 · 2026-04-12 21:24:30

648

Views

Prove that the following function is a norm

Published on 12 Apr 2026 - 21:24

score 1.5k · Answer 9 · 2026-04-12 11:55:56

1.5k

Views

Is the Frobenious norm of a matrix orthogonally invariant?

Published on 12 Apr 2026 - 11:55

score 48 · Answer 10 · 2026-04-16 23:07:20

48

Views

$M= \begin{bmatrix} 1 & -1 & 1\\ 2 & 1 & 2 \end{bmatrix}$,find a rank one $2 \times 3$ matrix $N$ such that $\|M-N\|_2$ is minimized

Published on 16 Apr 2026 - 23:07

score 237 · Answer 11 · 2026-04-16 02:23:40

237

Views

2-norm of upper non-singular $n \times n$ submatrix as an upper bound of the 2-norm of the entire $m\times n$ matrix

Published on 16 Apr 2026 - 2:23

score 147 · Answer 12 · 2026-04-15 12:28:15

147

Views

$\sup=\max$ norm in a normed space

Published on 15 Apr 2026 - 12:28

score 71 · Answer 13 · 2026-04-15 21:55:32

71

Views

$A^H=A^{-1}$ implies $\|x\|_2 = \|Ax\|_2$ for any $x\in \mathbb{C}^n$

Published on 15 Apr 2026 - 21:55

score 1.7k · Answer 14 · 2026-04-17 03:59:29

1.7k

Views

Sketch the open ball at the origin $(0,0)$, and radius $1$.

Published on 17 Apr 2026 - 3:59

score 475 · Answer 15 · 2026-04-15 17:48:47

How to prove that $\lVert x + y\rVert = \lVert x\rVert + \lVert y \rVert \implies \lVert tx + (1-t)y\rVert = t\lVert x\rVert + (1-t)\lVert y \rVert$?

List Question

Prove that the set $\{x\in \Bbb R^n:\|x\|_1=1\}$ is compact .

Prove that the set of all unit vectors in a finite-dimensional normed vector space is compact.

If $A$ is a continuous linear operator on $l^2$ such that $||A − T|| < 1$, show that $A$ is not invertible.

A question about uniformly bounded set of operators

Hyperplane separation theorem

Prove that the set of functions $\sin kx$ is closed and bounded

Counter example to show convergence in $l^\infty$ metric does not imply the same in $l^1$ metric on a specified metric space

Prove that the following function is a norm

Is the Frobenious norm of a matrix orthogonally invariant?

$M= \begin{bmatrix} 1 & -1 & 1\\ 2 & 1 & 2 \end{bmatrix}$,find a rank one $2 \times 3$ matrix $N$ such that $\|M-N\|_2$ is minimized

2-norm of upper non-singular $n \times n$ submatrix as an upper bound of the 2-norm of the entire $m\times n$ matrix

$\sup=\max$ norm in a normed space

$A^H=A^{-1}$ implies $\|x\|_2 = \|Ax\|_2$ for any $x\in \mathbb{C}^n$

Sketch the open ball at the origin $(0,0)$, and radius $1$.

How to prove that $\lVert x + y\rVert = \lVert x\rVert + \lVert y \rVert \implies \lVert tx + (1-t)y\rVert = t\lVert x\rVert + (1-t)\lVert y \rVert$?

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