$X$ is a normed space and for any $x \in X$ and $r \gt 0$, let $T=\{y\in X: \|y-x\|\le r\}$ and $S=\{y\in X: \|y-x\|\lt r\}$, then $\bar{S}=T$.
111830
When is the closure of an open ball equal to the closed ball?
155579
Show that $End_A(A)$ = {$r_a$ | $a ∈ A$}
50160
The quotient norm on $\ell^{\infty}(\mathbb{N}) / c_0 (\mathbb{N})$ is given by $\limsup_{n \in \mathbb{N}} |x_n|$.
60887
$d_A(x) = \inf_{a \in A} \|x-a\|$ is convex if and only if $A$ is convex
63104
Dual cone of $K = \{(x,t) \mid \| \boldsymbol{x} \|_1 \le t \}$
82334
For $f$ in dual space, there exists $x$ with norm 1 and $f(x)=\|f\|$ if space is reflexive (and nontrivial)
52095
Prove either $G=ST$ or |$G|\geq|S|+|T|$
73254
spectral radius in normed space is subadditive.
155884
What are the conditions on $\text{tr}(AB) \leq \text{tr(A)} \text{tr(B)}$ to be true?
60655
$\ker T\subset \ker S\Rightarrow S=rT$ when $S$ and $T$ are linear functionals
81229
Show that $(C([a,b]),\|{\cdot}\|_2)$ is not a complete normed vector space
14509
Contraction Map on Compact Normed Space has a Fixed Point
57868
Prove $\tau$ is cofinite on $X$ iff $(X,\tau)$ is $T_1$-space and any infinite subset of $X$ is dense in $(X, \tau)$
20335
If $A\subset B$ then $\bar{A}\subset \bar{B}$
51177
When are two norms equivalent on a Banach space?
148001
In Hilbert space: $x_n → x$ if and only if $x_n \to x$ weakly and $\Vert x_n \Vert → \Vert x \Vert$.
125776
If space of bounded operators L(V,W) is Banach, V nonzero, then W is Banach (note direction of implication)