Question

For any $f\in L^1(S^1)$ and let $y\in\mathbb{R}$, let $f_y$ be defined as $f_y(x)=f(x+y)$. Prove that $\lim_{y\to 0 } ||f_y-f||=0.$

score 84 · Answer 1 · 2018-02-11 16:09:11

84

Views

For any $f\in L^1(S^1)$ and let $y\in\mathbb{R}$, let $f_y$ be defined as $f_y(x)=f(x+y)$. Prove that $\lim_{y\to 0 } ||f_y-f||=0.$

Published on 11 Feb 2018 - 16:09

score 152 · Answer 2 · 2026-03-27 12:37:39

152

Views

Countable union covers a second countable, compact Hausdorff space

Published on 27 Mar 2026 - 12:37

score 80 · Answer 3 · 2026-03-25 08:51:30

80

Views

$\left\{e\right\}$ Closed Implies $T_0$ and Regular Topological Group

Published on 25 Mar 2026 - 8:51

score 5.3k · Answer 4 · 2026-03-27 14:27:08

5.3k

Views

Proof that the interior of any set is open

Published on 27 Mar 2026 - 14:27

score 466 · Answer 5 · 2018-02-11 23:53:20

466

Views

Collection of open, dense subsets of $X$ is a topology on $X$

Published on 11 Feb 2018 - 23:53

score 65 · Answer 6 · 2018-02-11 23:59:48

65

Views

Relationship between $ [0,1]^{n} $ and $ R^{n} $

Published on 11 Feb 2018 - 23:59

score 74 · Answer 7 · 2018-02-12 00:01:39

74

Views

Why should $2^{\aleph_0}$ be the maximum cardinality for regular spaces with countable bases?

Published on 12 Feb 2018 - 0:01

score 207 · Answer 8 · 2026-03-20 06:46:18

207

Views

Basic question regarding diffeomorphism and path connectedness

Published on 20 Mar 2026 - 6:46

score 71 · Answer 9 · 2026-03-22 12:34:24

71

Views

What is the hausdorff completion of this uniform structure on the real line?

Published on 22 Mar 2026 - 12:34

score 248 · Answer 10 · 2026-03-22 23:57:18

248

Views

Why does a zero-dimensional irreducible space have no non-trivial open subsets?

Published on 22 Mar 2026 - 23:57

score 39 · Answer 11 · 2018-02-12 07:37:32

39

Views

Prove the continuity of a correspondence

Published on 12 Feb 2018 - 7:37

score 199 · Answer 12 · 2018-02-12 13:06:11

199

Views

Problems on Metric Spaces

Published on 12 Feb 2018 - 13:06

score 28 · Answer 13 · 2018-02-12 13:16:42

Prove that if $(X,\theta)$ is a $T_{3} $ space and A is a dense subset of $X$, $\chi(a,A)=\chi(a,X),\forall a\in A$

score 4.3k · Answer 14 · 2026-03-26 18:57:46

4.3k

Views

Nested sequence of non-empty compact subsets - intersection differs from empty set

Published on 26 Mar 2026 - 18:57

score 48 · Answer 15 · 2026-03-26 18:56:56

48

Views

$T_2\implies T_4$ without compactness hypothesis?

Published on 26 Mar 2026 - 18:56

List Question

For any $f\in L^1(S^1)$ and let $y\in\mathbb{R}$, let $f_y$ be defined as $f_y(x)=f(x+y)$. Prove that $\lim_{y\to 0 } ||f_y-f||=0.$

Countable union covers a second countable, compact Hausdorff space

$\left\{e\right\}$ Closed Implies $T_0$ and Regular Topological Group

Proof that the interior of any set is open

Collection of open, dense subsets of $X$ is a topology on $X$

Relationship between $ [0,1]^{n} $ and $ R^{n} $

Why should $2^{\aleph_0}$ be the maximum cardinality for regular spaces with countable bases?

Basic question regarding diffeomorphism and path connectedness

What is the hausdorff completion of this uniform structure on the real line?

Why does a zero-dimensional irreducible space have no non-trivial open subsets?

Prove the continuity of a correspondence

Problems on Metric Spaces

Prove that if $(X,\theta)$ is a $T_{3} $ space and A is a dense subset of $X$, $\chi(a,A)=\chi(a,X),\forall a\in A$

Nested sequence of non-empty compact subsets - intersection differs from empty set

$T_2\implies T_4$ without compactness hypothesis?

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