User Norbert

117 votes

Accepted

Relations between p norms

answered Oct 21, 2012 at 16:45

63 votes

Accepted

Proving $\frac{\sin x}{x} =\left(1-\frac{x^2}{\pi^2}\right)\left(1-\frac{x^2}{2^2\pi^2}\right) \left(1-\frac{x^2}{3^2\pi^2}\right)\cdots$

sequences-and-series trigonometry infinite-product

answered Jun 12, 2012 at 13:58

44 votes

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How to show that quotient space $X/Y$ is complete when $X$ is Banach space, and $Y$ is a closed subspace of $X$?

functional-analysis vector-spaces banach-spaces

answered Mar 3, 2013 at 22:56

38 votes

$\ell^p\subseteq\ell^q$ for $0<p<q<\infty$ and $\|\cdot\|_q<\|\cdot\|_p$

real-analysis functional-analysis

answered Feb 29, 2012 at 1:43

34 votes

Accepted

Evaluating $ \lim\limits_{n\to\infty} \sum_{k=1}^{n^2} \frac{n}{n^2+k^2} $

sequences-and-series limits

answered Sep 4, 2012 at 14:25

31 votes

Surely You're Joking, Mr. Feynman! $\int_0^\infty\frac{\sin^2x}{x^2(1+x^2)}\,dx$

calculus real-analysis integration definite-integrals improper-integrals

answered May 21, 2014 at 12:41

28 votes

Prove that $X'$ is a Banach space

functional-analysis banach-spaces normed-spaces

answered Apr 14, 2012 at 20:19

28 votes

Gelfand Topology and C*-algebras

functional-analysis operator-algebras

answered Feb 17, 2013 at 22:32

27 votes

Accepted

Compute the limit of $\frac1{\sqrt{n}}\left(1^1 \cdot 2^2 \cdot3^3\cdots n^n\right)^{1/n^2}$

calculus real-analysis limits

answered Jun 23, 2012 at 20:30

26 votes

Accepted

Proving that closed (and open) balls are convex

functional-analysis

answered Apr 27, 2012 at 17:55

26 votes

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Properties of reflexive Banach spaces

general-topology functional-analysis hilbert-spaces banach-spaces big-list

answered Aug 4, 2014 at 16:07

25 votes

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Why is $L^1(\mathbb{R}^n) \cap L^2(\mathbb{R}^n)$ dense in $ L^2(\mathbb{R}^n)$?

real-analysis general-topology functional-analysis lp-spaces

answered Dec 27, 2012 at 20:36

24 votes

Non-closed subspace of a Banach space

banach-spaces

answered Aug 2, 2012 at 9:39

23 votes

Accepted

Why isometric isomorphic between Banach spaces means we can identify them?

functional-analysis banach-spaces

answered May 16, 2013 at 22:05

23 votes

Accepted

Inequality. $a^2+b^2+c^2 \geq a+b+c$

inequality

answered Aug 12, 2012 at 12:18

22 votes

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if $f$ is entire and $|f(z)| \leq 1+|z|^{1/2}$, why must $f$ be constant?

complex-analysis

answered May 30, 2012 at 19:13

22 votes

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Why study operator spaces?

functional-analysis operator-algebras operator-spaces

answered Sep 26, 2013 at 22:26

22 votes

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Why do we distinguish the continuous spectrum and the residual spectrum?

functional-analysis operator-theory spectral-theory

answered Jul 24, 2012 at 20:41

21 votes

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Weak Convergence implies boundedness and componentwise convergence

functional-analysis convergence-divergence hilbert-spaces

answered Oct 15, 2012 at 15:04

21 votes

Accepted

The diameter of a compact set.

real-analysis general-topology

answered Oct 7, 2012 at 22:23

20 votes

Evaluation of the limit $\lim\limits_{n \to \infty } \frac1{\sqrt n}\left(1 + \frac1{\sqrt 2 }+\frac1{\sqrt 3 }+\cdots+\frac1{\sqrt n } \right)$

calculus limits

answered Nov 16, 2012 at 16:38

20 votes

Maclaurin expansion of $\arcsin x$

calculus

answered Sep 17, 2012 at 5:18

20 votes

Evaluate $\int_{0}^{\infty} \frac{{(1+x)}^{-n}}{\log^2 x+\pi^2} \ dx, \space n\ge1$

calculus real-analysis integration improper-integrals

answered Oct 6, 2012 at 19:58

19 votes

Accepted

How to prove: $\left(\frac{2}{\sqrt{4-3\sqrt[4]{5}+2\sqrt[4]{25}-\sqrt[4]{125}}}-1\right)^{4}=5$?

calculus arithmetic radicals

answered Aug 10, 2014 at 8:18

18 votes

Calculating the shortest possible distance between points

calculus trigonometry derivatives

answered Jun 3, 2012 at 10:51

18 votes

Prove $(-a+b+c)(a-b+c)(a+b-c) \leq abc$, where $a, b$ and $c$ are positive real numbers

inequality

answered Jul 14, 2012 at 18:38

18 votes

How to evaluate $ \lim \limits_{n\to \infty} \sum \limits_ {k=1}^n \frac{k^n}{n^n}$?

calculus sequences-and-series

answered Jun 28, 2012 at 8:51

17 votes

Accepted

First theorem in Topological vector spaces.

functional-analysis topological-vector-spaces

answered Jun 7, 2012 at 17:40

16 votes

Accepted

Continuity of the adjoint map in various operator topologies

functional-analysis operator-theory

answered Aug 23, 2013 at 15:29

16 votes

Accepted

Distance between a point and a m-dimensional space in n-dimensional space ($m<n$)

linear-algebra geometry computer-science

answered Jul 22, 2012 at 15:30

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925 Answers

Relations between p norms

Proving $\frac{\sin x}{x} =\left(1-\frac{x^2}{\pi^2}\right)\left(1-\frac{x^2}{2^2\pi^2}\right) \left(1-\frac{x^2}{3^2\pi^2}\right)\cdots$

How to show that quotient space $X/Y$ is complete when $X$ is Banach space, and $Y$ is a closed subspace of $X$?

$\ell^p\subseteq\ell^q$ for $0<p<q<\infty$ and $\|\cdot\|_q<\|\cdot\|_p$

Evaluating $ \lim\limits_{n\to\infty} \sum_{k=1}^{n^2} \frac{n}{n^2+k^2} $

Surely You're Joking, Mr. Feynman! $\int_0^\infty\frac{\sin^2x}{x^2(1+x^2)}\,dx$

Prove that $X'$ is a Banach space

Gelfand Topology and C*-algebras

Compute the limit of $\frac1{\sqrt{n}}\left(1^1 \cdot 2^2 \cdot3^3\cdots n^n\right)^{1/n^2}$

Proving that closed (and open) balls are convex

Properties of reflexive Banach spaces

Why is $L^1(\mathbb{R}^n) \cap L^2(\mathbb{R}^n)$ dense in $ L^2(\mathbb{R}^n)$?

Non-closed subspace of a Banach space

Why isometric isomorphic between Banach spaces means we can identify them?

Inequality. $a^2+b^2+c^2 \geq a+b+c$

if $f$ is entire and $|f(z)| \leq 1+|z|^{1/2}$, why must $f$ be constant?

Why study operator spaces?

Why do we distinguish the continuous spectrum and the residual spectrum?

Weak Convergence implies boundedness and componentwise convergence

The diameter of a compact set.

Evaluation of the limit $\lim\limits_{n \to \infty } \frac1{\sqrt n}\left(1 + \frac1{\sqrt 2 }+\frac1{\sqrt 3 }+\cdots+\frac1{\sqrt n } \right)$

Maclaurin expansion of $\arcsin x$

Evaluate $\int_{0}^{\infty} \frac{{(1+x)}^{-n}}{\log^2 x+\pi^2} \ dx, \space n\ge1$

How to prove: $\left(\frac{2}{\sqrt{4-3\sqrt[4]{5}+2\sqrt[4]{25}-\sqrt[4]{125}}}-1\right)^{4}=5$?

Calculating the shortest possible distance between points

Prove $(-a+b+c)(a-b+c)(a+b-c) \leq abc$, where $a, b$ and $c$ are positive real numbers

How to evaluate $ \lim \limits_{n\to \infty} \sum \limits_ {k=1}^n \frac{k^n}{n^n}$?

First theorem in Topological vector spaces.

Continuity of the adjoint map in various operator topologies

Distance between a point and a m-dimensional space in n-dimensional space ($m<n$)