checking definition of bounded linear function involves operator maps between different spaces Let $H$ and $K$ be two Hilbert spaces. Let $T:K\to H$ be a bounded linear operator. Denote the inner products on $H$ and $K$ by $\langle\cdot,\cdot\rangle_H$, $\langle\cdot,\cdot\rangle_K$. Fix any $y\in H$; then the linear mapping 
$$ \Phi: K\to\mathbb{C},\quad\Phi x :=\langle Tx,y\rangle_H$$
defines a bounded functional on the Hilbert space $K$.
Could anyone help with the proof of above claim?
Recall, a functional simply a linear operator $\Phi:H\to \mathbb{C}$. 
$\Phi$ is bounded if there exists $K$ such that $|\Phi x|\le K\|x\|,\,\,\forall\,\,x\in H$. So I have
$$|\Phi x|=|\langle Tx,y\rangle_H|\le\cdots$$
Please help.

Hint: How do you prove that $y^*\colon H\to {\bf C}$ defined by $y^*(y'):=\langle y',y\rangle$ is bounded? Or, if you know what an adjoint operator is, just notice that $\Phi$ is just $T^*(y^*)$.
