For a bounded linear operator $\phi$, show that $||\phi(x)|| \leq ||\phi|| \cdot ||x||$, where $||\phi|| = sup_{||x|| = 1} ||\phi(x)||$
4498
Proof that $\|fx\| \leq \|f\|\cdot\|x\|$
144112
Show that $c_0$ is a Banach space with the norm $\rVert \cdot \lVert_\infty$
87897
Bounded operators on separable Hilbert spaces
89555
Inequality for the p norm of a convolution
101286
How show that $|a_{n}-1|\le c\lambda ^n,\lambda\in (0,1)$
124844
Show that $\phi \circ f$ belongs to $L_p$ for each $f \in L_p$
26183
Prob. 9, Sec. 4.3 in Kreyszig's Functional Analysis Book: Proof of the Hahn Banach Theorem without Zorn's Lemma
144656
Linear functional and a bounded norm
55067
Norm of self-adjoint operator
88687
Proof of the inequality $||a+b|^p - |a|^p| \leq \epsilon |a|^p + C_{\epsilon} |b|^p$
152887
Let $T : V \rightarrow V$ be a linear operator. If $T^n = O_V$ for some $n \ge 1$, prove that $I_V + T$ is an isomorphism.
3038
For positive invertible operators $C\leq T$ on a Hilbert space, does it follow that $T^{-1}\leq C^{-1}$?
50273
How to prove $\lvert \lVert x \rVert - \lVert y \rVert \rvert \overset{\heartsuit}{\leq} \lVert x-y \rVert$?
108781
$A$ and $B$ are bounded linear operators from the normed linear space $X$ to itself. If $AB$ is invertible are $A$ and $B$ invertible?
149997
Convolution operator has norm 1 with respect to $L^2$ norm
90435
Prove that there exist linear functionals $L_1, L_2$ on $X$
74501
$\delta : C([0,1])\to\mathbb{R}$, $f\mapsto f(0)$, with sup-norm is bounded but unbounded with one-norm.