Let $H,K$ be Hilbert spaces with $h\in H,k\in K$. Let $k\otimes h^\ast(g)= \left\langle g,h \right\rangle k$. I'm supposed to prove $ (k\otimes h^\ast)^\ast=h\otimes k^\ast$, but I don't see how this makes any sense, since
$$\left\langle k\otimes h^\ast (x),y \right\rangle =\left\langle \left\langle x,h \right\rangle k,y \right\rangle = \left\langle k,\left\langle h,x \right\rangle y \right\rangle $$ and
$$ \left\langle x,h\otimes k^\ast (y) \right\rangle = \left\langle x, \left\langle y,k \right\rangle ,h \right\rangle = \left\langle \left\langle k,y \right\rangle x,h \right\rangle $$ which do not seem even remotely the same...