Question

Let $X=(C[0,1],||\cdot||_\infty)$ and $A=\{f\in X | f(0) \neq 0\}$. How can we decide whether $A$ is open or closed in $X$?

score 22 · Answer 1 · 2019-09-19 14:59:43

Let $X=(C[0,1],||\cdot||_\infty)$ and $A=\{f\in X | f(0) \neq 0\}$. How can we decide whether $A$ is open or closed in $X$?

score 156 · Answer 2 · 2019-09-19 15:44:05

156

Views

Show that $B_1^{d_1}(0) $ is open in $(C[0,1], d_\infty)$ but $B_1^{d_\infty}(0)$ is not open in $(C[0,1],d_1)$

Published on 19 Sep 2019 - 15:44

score 35 · Answer 3 · 2019-09-19 17:51:34

35

Views

$A=\{e_k : k=1,2,3\ldots\} $ is closed in $\ell1$

Published on 19 Sep 2019 - 17:51

score 37 · Answer 4 · 2019-09-19 18:17:23

37

Views

Is $c_{00} $ open or closed in $\ell^\infty$

Published on 19 Sep 2019 - 18:17

score 61 · Answer 5 · 2019-09-19 19:10:54

61

Views

$x_{n} \rightarrow x$ and $f(x_{n}) \rightarrow f(x)$ is $f$ continuous?

Published on 19 Sep 2019 - 19:10

score 123 · Answer 6 · 2019-09-19 20:46:29

123

Views

What are the interior points of this set $A=\{(x_n)\in \ell_2 \mid |x_n|<\frac1n, n=1,2,3\ldots\}$?

Published on 19 Sep 2019 - 20:46

score 125 · Answer 7 · 2026-04-11 15:25:46

125

Views

A metric on $\mathbb R^2$

Published on 11 Apr 2026 - 15:25

score 191 · Answer 8 · 2026-04-12 03:34:49

191

Views

Show that the sum of two uniformly continuous functions is uniformly continuous in an arbitrary metric space

Published on 12 Apr 2026 - 3:34

score 227 · Answer 9 · 2019-09-20 03:17:47

227

Views

Prove that $f$ is a homeomorphism

Published on 20 Sep 2019 - 3:17

score 367 · Answer 10 · 2019-09-20 03:28:24

367

Views

$f$ is continuous relative to $A$ at, 2

Published on 20 Sep 2019 - 3:28

score 41 · Answer 11 · 2019-09-20 06:13:48

41

Views

Construction of a continuous function on $[0, 1]$

Published on 20 Sep 2019 - 6:13

score 376 · Answer 12 · 2026-04-09 09:55:56

376

Views

Relative compact sequence

Published on 09 Apr 2026 - 9:55

score 75 · Answer 13 · 2019-09-20 16:59:19

75

Views

Let $A$ and $B$ be subsets of $M$ and let $f: M \rightarrow \mathbb{R}$. Prove or disprove the following statements:

Published on 20 Sep 2019 - 16:59

score 254 · Answer 14 · 2026-04-09 06:06:38

254

Views

Existence of increasing sequence of compact sets

Published on 09 Apr 2026 - 6:06

score 163 · Answer 15 · 2026-04-01 00:27:16

163

Views

Prove that $f_n \rightharpoonup f$ iff $\sup_n ||f_n||_{L^{p}} < \infty$ in $p \in (1, \infty), f_n,f \in L^p (0,1)$

Published on 01 Apr 2026 - 0:27

List Question

Let $X=(C[0,1],||\cdot||_\infty)$ and $A=\{f\in X | f(0) \neq 0\}$. How can we decide whether $A$ is open or closed in $X$?

Show that $B_1^{d_1}(0) $ is open in $(C[0,1], d_\infty)$ but $B_1^{d_\infty}(0)$ is not open in $(C[0,1],d_1)$

$A=\{e_k : k=1,2,3\ldots\} $ is closed in $\ell1$

Is $c_{00} $ open or closed in $\ell^\infty$

$x_{n} \rightarrow x$ and $f(x_{n}) \rightarrow f(x)$ is $f$ continuous?

What are the interior points of this set $A=\{(x_n)\in \ell_2 \mid |x_n|<\frac1n, n=1,2,3\ldots\}$?

A metric on $\mathbb R^2$

Show that the sum of two uniformly continuous functions is uniformly continuous in an arbitrary metric space

Prove that $f$ is a homeomorphism

$f$ is continuous relative to $A$ at, 2

Construction of a continuous function on $[0, 1]$

Relative compact sequence

Let $A$ and $B$ be subsets of $M$ and let $f: M \rightarrow \mathbb{R}$. Prove or disprove the following statements:

Existence of increasing sequence of compact sets

Prove that $f_n \rightharpoonup f$ iff $\sup_n ||f_n||_{L^{p}} < \infty$ in $p \in (1, \infty), f_n,f \in L^p (0,1)$

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