Fredolm theory in A. Friedman's book

Question

The image below is part of Fredholm theory in A. Friedman's book. This is to prove $$\dim(\ker(\lambda-T))=\dim(\ker(\lambda-T)^*).$$ I don't understand how to reach the highlighted part: $\lambda x-Tx=0$

Could anybody help? Thanks.

Answer 1

$\lambda x-T(x)=\sum_{i=1}^{i=n} x_i^*(x)y_i$. It is an element of $Vect(y_1,...,y_n)$ The author shows that $x_i^*(x)=0, i=1,...,n$ this implies that $\lambda x-T(x)=0$ since its coordinates are zero in the basis $(y_1,...,y_n)$ of $Vect(y_1,...,y_n)$.