Applying the definition of adjoint operator

Question

How do I apply the definition of adjoint operator in this problem? U and V are two arbitrary operators, not necessarily Hermitian. Show that (UV )† = V †U†.

Answer 1

We have $$\langle (UV)^Tx, \, y\rangle = \langle x,\, UVy\rangle =\langle U^Tx, \, Vy\rangle =\langle V^T(U^Tx), \, y\rangle$$ Substituting the standard basis vectors $e_i$ and $e_j$ for $x$ and $y$ yields that the $i,j$ entries of $(UV)^T$ and $V^TU^T$ are the same.