An Hilbert space problem

Question

I have a problem with this problem, could anyone help me? thankyou very much!

Let $H_1$ and $H_2$ be Hilbert spaces, and let $x\in H_1$, $x\ne 0$ and $y\in H_2$. a) show there is a functional $f\in H_1$ such that $\|f\|=1$ and $f(x)=\|x\|$

b) build an operator $T\in L(H_1,H_2)$ such that $Tx=y$.

For a), I have considered $f(y)=\frac{1}{\|x\|}\langle y,x\rangle$ (I have update an image beacause I can´t write on LaTex here)

For b) I have no idea for it, could anyone help me?

Answer 1

let $f_1:H_1\rightarrow \mathbb{C}$ be the functional you found in part (a). you can define an operator $T:H_1\rightarrow H_2$ by the formula: $$\forall z \in H_1:T(z)=||x||^{-1}f_1(z)y$$ Its easy to check this operator is linear and its bounded because if $||z|| =1$ then: $$ ||T(z)||=||x||^{-1}||y||\cdot |f_1(z)|\leq \frac{||f_1||\cdot||y||}{||x||}$$

And clearly T(x)= y