Let $H$ be hilbert and $T$ a BLO, such that $T:H\rightarrow H$. Prove that $\langle T(x),x \rangle = 0$ implies $T = 0$.
11715
Prove that if $\langle Tx,x\rangle =0$ for all $x \in X$, then $T = 0$
76790
Condition to guarantee $f=0$ on $[a,b]$
101861
How do you prove that $tr(B^{T} A )$ is a inner product?
30996
If $\langle Ta, a\rangle \in \mathbb{R}$ for all $a$ then $T$ is self-adjoint
83183
Why is $\langle x| x' \rangle=\delta(x-x')$?
150199
How to prove for a finite group that $|\langle g \rangle|=o(g)$?
71064
How to prove that $\langle Av,w\rangle=\langle v,A^Tw\rangle$?
71187
How to prove $\hat{p}|x\rangle=i\hbar\frac{\partial}{\partial x}|x\rangle$,using $[\hat{x},\hat{p}]=i\hbar$?
65395
Prove that if $\operatorname{rank}(T) = \operatorname{rank}(T^2)$ then $R(T) \cap N(T) = \{0\}$
110545
Space $X$ such that $H_i(X,\mathbb{Z})\neq 0$ but $H_i(X,\mathbb{Z}_3)= 0$
88442
Prove that there's no ordered basis $E$ in which $T{x\choose y}={0\choose y}$ can be represented as $1\ 2\choose 2\ 4$
48238
Prove the positive definiteness of Hilbert matrix
103839
Proving that $T^{2} = T$ for a linear operator on $W$ implies that $V = \operatorname{Null} T \oplus \operatorname{Range} T$
38584
$T\colon L^2[0,1] \to L^2[0,1]$ be a bounded linear map of Hilbert spaces such that if $f\in L^2[0,1]$ is continuous then so is $Tf$.
2105
$\langle Tu\;|\;u\rangle=0,\;\forall u\in E \Longrightarrow T=0$?
144225
$\ell_p$ is Hilbert if and only if $p=2$
56818
Prove that if $A$ is positive, then $\exists\;\alpha>0$ such that $\langle Ax, x \rangle\geq \alpha|x|^2$ for $x\in\Bbb{R}^n$