Suppose that A is a linear map from a Hilbert space H into itself that satisfies  =  for all x, y in H . Show that A is bounded.
52198
Why is every selfadjoint operator closed?
12212
Showing a certain subspace of Hilbert space is dense
135157
Totally bounded, complete $\implies$ compact
9451
Show bounded and convex function on $\mathbb R$ is constant
55461
How to prove that if $a$ belongs to $\mathbb R$, such that $0\leq a \leq\epsilon$, then $a = 0$
101776
Bounded solution for an equation
86481
Let $H$ be an infinite-dimensional separable Hilbert space, show that a Hilbert-Schmidt operator is compact
118134
If $\psi h$ is in $L^2$ for all $h\in L^2$, must $\psi$ be essentially bounded?
148001
In Hilbert space: $x_n → x$ if and only if $x_n \to x$ weakly and $\Vert x_n \Vert → \Vert x \Vert$.
91022
How do I prove that a subspace of a vector space $X$ is the null space of some linear functional on $X$?
120648
Proof Check: Every Cauchy Sequence is Bounded
29836
Prove: Convergent sequences are bounded
17165
Is total boundedness a topological property?
45252
"Every linear mapping on a finite dimensional space is continuous"
130045
Show that $l^2$ is a Hilbert space
154226
Upper bound for norm of Hilbert space operator
119858
Proof: Every convergent sequence of real numbers is bounded