When my lecture notes discuss the adjoint of an operator in Banach spaces, it is defined like this. The adjoint of an operator [tex]T:X\to Y[/tex] is an operator [tex]T^*:Y^*\to X^*[/tex] so that for all [itex]f\in Y^*[/itex] and [itex]x\in X[/itex] [tex](T^* f)(x) = f(T x)[/tex]. But we get into Hilbert spaces, it is said to be given by the equation [tex](Tf|g) = (f|T^*g)[/tex] The Hilbert space is also a Banach space, so these definitions seem to be contradicting. In fact my lecture notes are unfortunately messy. I cannot tell for sure what precisely are the definitions, but this is what it says, approximately. Any major misunderstandings could be pointed out. I can conclude that I'm understanding something wrong, because I'm not understanding what the adjoint really is.