Let $V$,$W$ be finite dimensional vector spaces and let $f$ be a linear map,
define $f^*$ to be the adjoint of $f$.
If $S$,$T$ are tensors on $W$ show that $f^∗(S\otimes T) = (f^∗S)\otimes (f^∗T)$
What I have so far is:
$f^∗(S\otimes T)(v_1,...,v_{2k}) = (S\otimes T)(f(v_1),...,f(v_{2k})) =S(f(v_1),...,f(v_k))T(f(v_{k+1}),...,f(v_{2k})) =f^*S(v_1,...,v_k)f^*T(v_{k+1},...,v_{2k})$
I'm not sure what step to take next to make it a tensor product, the textbook I'm using said its easy, but I don't know what I'm missing