Im having some difficulties proving some basic properties of the adjoint operator. I want to prove the following things: 1) There exists a unique map [itex]T^*:K\rightarrow H[/itex] 2) That [itex]T^*[/itex] is bounded and linear. 3) That [itex]T\rightarrow K[/itex] is isometric if and only if [itex]T^*T = I[/itex]. 4) Deduce that if [itex]T[/itex] is an isometry, then [itex]T[/itex] has closed range. 5) If [itex]S \in B(K,H)[/itex], then [itex](TS)^* = S^*T^*[/itex], and that [itex]T^*^* = T[/itex]. 6) Deduce that if [itex]T[/itex] is an isometry, then [itex]TT^*[/itex] is the projection onto the range of [itex]T[/itex]. Note that [itex]H,K[/itex] are Hilbert Spaces. There are quite a few questions, and I am hoping that by proving each one I will get a much better understanding of these adjoint operators. Now I think I have made a fairly good start with these proofs, so I'd like someone to check them please. We'll begin with the first one.