Is $f_{y}$ a bounded linear map?

Question

Let $T \in BL(H)$ where $H$ is a Hilbert space

For fixed $y \in H$ , define $f_{y} : H \rightarrow K$ by $f_{y} = \langle Tx,y\rangle$

Is $f_{y}$ a bounded linear map on $H $ ?

So i tried like this -

$|f_{y}(x)| = |\langle Tx,y\rangle| \leq ||Tx|| . ||y|| \leq ||T|| . ||x||. ||y|| $

(Above by Cauchy Schwarz Inequality)

Now $||f_{y}|| = \sup\{|f_{y}(x)| : ||x|| \leq 1\}$

so $||f_{y}|| \leq ||T|| . ||y||$ so it is bounded on $H$.

Is this correct?. At the same time I think that it is wrong as to show $f_{y}$ is bounded on $H$ I have to show $|f_{y}(x)| \leq M\|x\|\ \forall x \in H$.

EDIT -

Perhaps it will be over if I take $||T|| ||y|| =M$ thus we get a bounded linear map as then $|f_{y}(x)| \leq M ||x|| \forall x \in H$

Answer 1

Yes,as per the edit, if I take $||T|| ||y|| =M$ thus we get a bounded linear map

as then $|f_{y}(x)| \leq M ||x|| \forall x \in H$ which we require!