Why $\sup_{\substack{x\in E,\\ \|x\|=1}}\|Ax\|\geq\sup_{\substack{x\in E,\\ \|x\|\leq1}}\|Ax\|?$
62425
Equivalent Definitions of the Operator Norm
99165
How to prove that for Brownian motion in $(a, b)$ $\mathbb{E}^x[\min(H_a, H_b)] = (x-a)(b-x)$?
117257
Proving the supremum of the set {$\cos (1/x)$} is 1
50273
How to prove $\lvert \lVert x \rVert - \lVert y \rVert \rvert \overset{\heartsuit}{\leq} \lVert x-y \rVert$?
130207
Why is the p-value written with a supremum?
16779
Why does $\left\| {\left| A \right|} \right\| \le \left\| {\left| I \right|} \right\|$, for every doubly stochastic matrix $A \in M_n$?
19823
Inequality involving $\limsup$ and $\liminf$: $ \liminf(a_{n+1}/a_n) \le \liminf((a_n)^{(1/n)}) \le \limsup((a_n)^{(1/n)}) \le \limsup(a_{n+1}/a_n)$
155884
What are the conditions on $\text{tr}(AB) \leq \text{tr(A)} \text{tr(B)}$ to be true?
105107
How to explain the following definition of $\lim\sup s_n$ intuitively?
78754
Showing that $E[X|X<x]$ is smaller or equal than $E[X]$ for all x
142360
How to show $x_1,x_2, \dots ,x_n \geq 0 $ and $ x_1 + x_2 + \dots + x_n \leq \frac{1}{2} \implies (1-x_1)(1-x_2) \cdots (1-x_n) \geq \frac{1}{2}$
93157
If $A \subseteq B$, does $\mbox{dist}(x,\partial A) \le \mbox{dist}(x, \partial B)$ hold for all $x \in A$
125209
Prove $e^x - e^y \leq e |x-y|$ for $x$ belonging to $[0,1]$
50160
The quotient norm on $\ell^{\infty}(\mathbb{N}) / c_0 (\mathbb{N})$ is given by $\limsup_{n \in \mathbb{N}} |x_n|$.
113778
Show that $E=\{(x,\alpha)\mid 0\leq \alpha<|f(x)|\}$ is measurable if $f$ is measurable.
116768
Problem 5, sec. 13 in Munkres' Topology, 2nd ed.: How to prove the assertion if $\mathscr{A}$ is a subbasis?
46746
How to prove if $u\in W^{1,p}$, then $|u|\in W^{1,p}$?