Show that if $\langle Tx,y \rangle = 0$ for all $x,y \in X$, then $T$ is the zero operator.
11715
Prove that if $\langle Tx,x\rangle =0$ for all $x \in X$, then $T = 0$
14161
Proof of when is $A=X^TX$ invertible?
127047
How to show that $\sum_{i=1}^n | \langle f, f_i\rangle |^2 \leq \Vert f \Vert^2$
30569
Prove that the only operator on $\mathbb{C}$ for which his inner product is zero is zero
69515
If $x_n\cdot y_n\to 0$ then $x_n \to 0$ or $y_n \to 0 $
87188
Prove that a square-integrable entire function is identically zero
10001
Prove that the only eigenvalue of a nilpotent operator is 0?
111550
Let $A$ be a square complex matrix of size $n$ then $\langle Ax,x\rangle=0$ for $\forall x \in \Bbb C^n$ $\iff A=0$
32606
Is the operator $T((x_1,x_2,...,x_n))=(x_1,\frac{x_2}2,\frac{x_3}3,...)$ on $\ell^2$ self-adjoint and unitary
158343
how to prove an element is non-zero in a tensor-product
16789
Prove that $T^2$ is the zero transformation if and only if $\mathrm{range}(T) \subseteq \ker(T)$.
90345
If $\lVert x\rVert=\lVert y \rVert$, prove that exists a unitary transformation such as $Tx=y$
3682
If $V = \text{null}(\textsf{T}-\lambda\textsf{I}) \oplus \text{range}(\textsf{T}-\lambda\textsf{I})$, then prove that $\textsf{T}$ is diagonalizable
73406
Prove that $E_{\lambda}(T)=E_{\lambda^{-1}}(T^{-1})$ if $T$ is an invertible linear operator on a finite dimensional vector space $V$
102990
Prove or disprove: $S_{10} = \langle (1,3),(1,2, ... ,10) \rangle$
121987
Direct proof that nilpotent matrix has zero trace
145161
Testing whether an element of a tensor product of modules is zero