Question

Does $\lim\limits_{m\rightarrow\infty}\sum\limits_{k=m}^\infty d(a_k,a_{k+1})$ equal to zero for any sequence $(a_n)\subset M$

score 30 · Answer 1 · 2019-01-19 10:44:55

30

Views

Does $\lim\limits_{m\rightarrow\infty}\sum\limits_{k=m}^\infty d(a_k,a_{k+1})$ equal to zero for any sequence $(a_n)\subset M$

Published on 19 Jan 2019 - 10:44

score 20 · Answer 2 · 2019-01-19 16:41:27

20

Views

$\{a\}\subset M$ is open $\forall a\in M$ iff $(M,d)$ is discrete

Published on 19 Jan 2019 - 16:41

score 190 · Answer 3 · 2026-02-23 04:59:06

190

Views

Upper or lower bound of Wasserstein-2 metric on Gaussian distribution

Published on 23 Feb 2026 - 4:59

score 31 · Answer 4 · 2026-03-25 16:49:07

31

Views

In a complex metrizable topological vector space, $d(0,\alpha x)\neq |\alpha|d(0,x), \ \alpha \in \mathbb C.$

Published on 25 Mar 2026 - 16:49

score 124 · Answer 5 · 2026-03-25 22:26:10

124

Views

Subsets of complex plane with euclidean metric that are complete

Published on 25 Mar 2026 - 22:26

score 67 · Answer 6 · 2026-04-01 09:54:07

67

Views

Closure of ball in normed space

Published on 01 Apr 2026 - 9:54

score 370 · Answer 7 · 2026-03-28 07:58:49

370

Views

$\Bbb R^d$ with norm is a Banach

Published on 28 Mar 2026 - 7:58

score 95 · Answer 8 · 2019-01-20 18:14:58

95

Views

Can two closed disjoint subsets be at zero distance from each other?

Published on 20 Jan 2019 - 18:14

score 105 · Answer 9 · 2019-01-20 21:36:39

105

Views

Proof verification: Show that $f$ is continuous if $f(\overline{A})\subset\overline{f(A)}$.

Published on 20 Jan 2019 - 21:36

score 31 · Answer 10 · 2026-04-03 22:49:07

31

Views

Counting Balls in $L^2_m[0,1]$

Published on 03 Apr 2026 - 22:49

score 64 · Answer 11 · 2026-03-30 13:43:40

64

Views

Isometry $f:M\mapsto M$ is surjective if $M$ is compact: proof by composing $f$ with itself

Published on 30 Mar 2026 - 13:43

score 110 · Answer 12 · 2026-03-28 07:51:39

110

Views

Closure of finite span has non-empty interior

Published on 28 Mar 2026 - 7:51

score 189 · Answer 13 · 2026-03-26 08:02:39

189

Views

Showing that $f:M\mapsto M$ has a unique fixed point if $M$ is compact and $d(f(a),f(b))<d(a,b)~\forall a,b\in M$

Published on 26 Mar 2026 - 8:02

score 52 · Answer 14 · 2026-03-26 09:42:29

52

Views

$d(0,\alpha x+\beta y) \leq |\alpha|d(0,x)+|\beta|d(0,y);\ |\alpha|+|\beta|\leq 1$

Published on 26 Mar 2026 - 9:42

score 30 · Answer 15 · 2026-04-04 16:18:08

30

Views

proof of totally bounded and complete $\implies$ compact ($1/2^n$ vs $1/n$ radius open balls)

Published on 04 Apr 2026 - 16:18

List Question

Does $\lim\limits_{m\rightarrow\infty}\sum\limits_{k=m}^\infty d(a_k,a_{k+1})$ equal to zero for any sequence $(a_n)\subset M$

$\{a\}\subset M$ is open $\forall a\in M$ iff $(M,d)$ is discrete

Upper or lower bound of Wasserstein-2 metric on Gaussian distribution

In a complex metrizable topological vector space, $d(0,\alpha x)\neq |\alpha|d(0,x), \ \alpha \in \mathbb C.$

Subsets of complex plane with euclidean metric that are complete

Closure of ball in normed space

$\Bbb R^d$ with norm is a Banach

Can two closed disjoint subsets be at zero distance from each other?

Proof verification: Show that $f$ is continuous if $f(\overline{A})\subset\overline{f(A)}$.

Counting Balls in $L^2_m[0,1]$

Isometry $f:M\mapsto M$ is surjective if $M$ is compact: proof by composing $f$ with itself

Closure of finite span has non-empty interior

Showing that $f:M\mapsto M$ has a unique fixed point if $M$ is compact and $d(f(a),f(b))<d(a,b)~\forall a,b\in M$

$d(0,\alpha x+\beta y) \leq |\alpha|d(0,x)+|\beta|d(0,y);\ |\alpha|+|\beta|\leq 1$

proof of totally bounded and complete $\implies$ compact ($1/2^n$ vs $1/n$ radius open balls)

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