Show that the operator $T:L^2[0,1]\times L^2[0,1]$ defined by $(Th)(x)=\int_0^1L(x,y)h(y)dy$ is a finite rank operator

Question

Let $f_1, g_1,..., f_n, g_n \in C[0,1]$. Denote $L(x,y)=\sum_{i=1}^nf_i(x)g_i(x)$. How do we show that the operator $T:L^2[0,1]\times L^2[0,1]$ defined by $$(Th)(x)=\int_0^1L(x,y)h(y)dy$$ is a finite rank operator?

Answer 1

$(Th)(x) = \sum_k ( \int_0^1 g_k(y) h(y) dy )\ f_k(x) $, so $Th \in \operatorname{sp} \{ f_k \}_k$.