Need a help in understanding theorem 6.1 in chapter 2 in Israel Gohberg.

Question

The theorem and its proof is given in the following pictures:

But I could not understand:

$Q_{1}$ the line after equation(1), why $f_{i}(x)$ is given by the indicated form for all i?, and why it is a bounded linear functional?

$Q_{2}$ why the representation of $K$ is not unique?

Could anyone explain this for me please?

Answer 1

$f_{i}(x)=\left<Kx,\varphi_{i}\right>$ is simply by definition. To see that it is bounded, we have $|f_{i}(x)|\leq\|Kx\|\|\varphi_{i}\|=\|Kx\|\leq\|K\|\|x\|$. Note that $K$ is bounded by assumption.