On a normed space if $\|u+v\|=\|u\|+\|v\|$ then $\forall s,t \geqslant 0$ we have that $\|su+tv\|=s\|u\|+t\|v\|$
2315
How to prove that $\lVert x + y\rVert = \lVert x\rVert + \lVert y \rVert \implies \lVert tx + (1-t)y\rVert = t\lVert x\rVert + (1-t)\lVert y \rVert$?
132869
Show that any bounded linear functional on a normed linear space is continuous
132595
$\sup_{0 \leq M \leq I_{\mathcal{H}_1}} \frac{\operatorname{Tr}[\rho M]}{\operatorname{Tr}[\sigma M]}$ satisfies Data Processing Inequality
53633
Given two subspaces $U,W$ of vector space $V$, how to show that $\dim(U)+\dim(W)=\dim(U+W)+\dim(U\cap W)$
81229
Show that $(C([a,b]),\|{\cdot}\|_2)$ is not a complete normed vector space
154226
Upper bound for norm of Hilbert space operator
131609
Prove that $Au\cdot v = u\cdot A^Tv$
34173
Banach spaces and their unit sphere
74623
Associative rule of subspaces and vector spaces: $(U+V) \cap W = U + (V \cap W)$
109632
Show reflexive normed vector space is a Banach space
97158
Prove that the operator norm is a norm
158706
How convergence relates to equivalence of norms
150720
Given that $W$ is a subspace of $V$, prove: $\dim W=\dim V \implies W=V$
55067
Norm of self-adjoint operator
50160
The quotient norm on $\ell^{\infty}(\mathbb{N}) / c_0 (\mathbb{N})$ is given by $\limsup_{n \in \mathbb{N}} |x_n|$.
91022
How do I prove that a subspace of a vector space $X$ is the null space of some linear functional on $X$?
132487
Proving that if $V$ is an inner product space and $U_1, U_2$ are subsets, then $U_1 \subset U_2$ if and only if $U^{\perp}_2 \subset U^{\perp}_1$