this is a question on Banach Spaces that I have encountered. I am new to this type of stuff, and despite doing many exercise questions, I am still unsure where to begin.
Let $T\colon H\mapsto H$ be a bound linear operator on a Hilbert Space H. Meaning T is linear and satisfies the relation $||Tg||\le K||g|| \forall g\in H$ Show there exist a unique bounded mapping $T^*\colon H\mapsto H \ni (Tf,g)=(f,T^*g)\forall f,g\in H$.
If I'm not mistaken, $T^*$ is an adjoint of T. Since we're working with Hilbert Spaces, I'm thinking of applying one of the three Riesz Representation Theorems, perhaps the first one.
Any help is appreciated.
$f \to \langle Tf, g \rangle$ is a continuous linear map so (by Riesz Theorem) there exists an element $T^{*}g$ such that $f \to \langle Tf, g \rangle =\langle f, T^{*}g \rangle$. Linearilty of $T^{*}$ is easy to verify. Note that $|\langle f, T^{*}g \rangle |=| \langle Tf, g \rangle|\leq ||Tf|| \, ||g||\leq ||T||\, ||f||\, ||g||$. Taking sup over $f$ we get $||T^{*}g|| \leq ||T|| \, ||g||$ so $T^{*}$ is bounded.