$\displaystyle (H, \langle .,. \rangle)$ Hilbert space, and $\displaystyle f_1$,$\displaystyle f_2$ $\displaystyle \in H$.

Prove: $\displaystyle Af= \langle f,f_1 \rangle f_1 + \langle f,f_2 \rangle f_2$ is bounded linear operator, and calculate his norm.

...Idea: $\displaystyle |Af| \leq ||f|| ||f_1|| ||f_1|| + ||f|| ||f_2|| ||f_2||$ (??)