We have Hilbert Space $H$ with real inner product $H_1$ i.e $H,\langle .,. \rangle_{H_1}$ and there is another inner product $H_2$ on $H$ such that
$$\langle x,x \rangle_{H_1} \le \langle x,x \rangle _{H_2}$$
Now, let fix $ p \in H$ and define a map $T_p : H \to \mathbb R$ defined as $T_p(x)=\langle x,p \rangle_{H_1}$
I want to show $T_p$ is bounded linear functional on $H$. My attempt is following:
We say linear functional is bounded if $\exists c$ such that $||T_p(x)||_ {\mathbb R} \le c||x||_{H_1} $.
I had a problem to show this since we have two inner product. We know that norm on $\mathbb R$ is absolute value so $$|T_p (x)| = \langle x,p \rangle_{H_1} \le ||x||_ {H_1} ||p||_{H_1} $$
Last transition is from Cauchy-Schwarz then I said it is bounded but I am not sure about it.!!!!
By Riesz Representation Theorem we know that since $T_p$ is bounded (assume I showed it) then there is a $q \in H$ such that $$\langle x,p \rangle_{H_1} \le \langle x,q \rangle _{H_2}$$
Now, my question is how can I say the map $T: H \to H$ defined as $p \to q$ is linear where $p,q$ stated above. I stucked with this because it doesn't look like $T_p$(I guess). What should I do?