How do I show that in a normed space $| (\|x\|-\|y\|) | \leq \|x-y\|$
3820
Prove that $||x|-|y||\le |x-y|$
81618
Prove $||a|-|b| | \leq |a-b|$
51267
Showing a generalized version of the Hölder inequality for $L^{p}$ spaces
67313
Proving $\{(x_1, x_2) \in \mathbb{R}^2_{+} \mid x_1 x_2 \geq 1\}$ is convex
37924
Equivalence of norms proof
57096
Show that $\max{\{|a|+|b|,|c|+|d|\}} \leq \max{\{|a|,|c|\}}+\max{\{|b|,|d|\}}.$
144112
Show that $c_0$ is a Banach space with the norm $\rVert \cdot \lVert_\infty$
154762
Dual space of a finite dimensional normed space
125209
Prove $e^x - e^y \leq e |x-y|$ for $x$ belonging to $[0,1]$
31846
Does $\neg(x > y)$ imply that $y \geq x$?
25017
Why is every $p$-norm convex?
10383
The $ l^{\infty} $-norm is equal to the limit of the $ l^{p} $-norms.
51177
When are two norms equivalent on a Banach space?
22689
Reference for trace/norm inequality
17164
How do you show monotonicity of the $\ell^p$ norms?
45370
Is it true that $|d(x, z) - d(y, z)| \leq d(x, y)$ and how do I prove it?
49240
$\forall x,y \in \mathbb{R}:\vert\cos x- \cos y\vert\leq\vert x-y\vert$