Show that $A$ is self-adjoint if and only if $(Ax,x)$ is real for all x
30996
If $\langle Ta, a\rangle \in \mathbb{R}$ for all $a$ then $T$ is self-adjoint
90757
$T$ self-adjoint on Hilbert space $H$, $T^n=T^{n+1}$ will $T$ be an orthogonal projection operator?
35874
Show that when $BA = I$, the solution of $Ax=b$ is unique
95481
The set of self-adjoint operators over a Hilbert space doesn't form a lattice
28565
Prove $\operatorname{rank}A^TA=\operatorname{rank}A$ for any $A\in M_{m \times n}$
142725
The general solution of $x^a = a^x$ for real $a >0$
55488
Proving that $\operatorname{codim}(X,Y)=\dim(\mathcal{O}_{X,\eta})$
100845
How to prove that $G(x)=ax+b$ is a one-to-one correspondence where $a\neq0$ and $a,b\in\mathbb{R}$.
27248
If $A$ is an $m \times n$ matrix, prove that $Ax=0$ has infinite solutions $\iff$ $\text{rank}(A)<n$
43913
Prove that if $A^*+A=AA^*$ then $A$ is normal
54668
Proof for the completeness of eigenfunctions of a self-adjoint operator
118996
Prove that $T$ is an orthogonal projection
57609
Hahn-Banach proof by extension of basis
105852
How to verify whether or not there exists a vector $x$ such that $Ax > 0$?
60723
Prove that if $A$ is normal, then eigenvectors corresponding to distinct eigenvalues are necessarily orthogonal (alternative proof)
155579
Show that $End_A(A)$ = {$r_a$ | $a ∈ A$}
118049
Show that $\mathbb{R}^{\mathbb{R}} = U_{e} \oplus U_{o}$