Exercise 16, § 1

Let $X = [0,1]$ endowed with Lebesgue measure.

a) For every sequence $(f_n)_{n\in\mathbf{N}}$ of continuous functions on X which is orthonormal in $L^2_{\mathbf{C}}$(X), there exists a kernel $k\in \mathscr{C}(X\times X)$ such that

a) The endomorphism $u_k$ of $L^2_{\mathbf{C}}(X)$ is positive ;

b) For every $n\in \mathbf{N}$, the function $f_n$ is a proper function of $u_k$ whose eigenvalue is non-zero ;

c) The family $(f_n)$ is a basis of the image of the endomorphism $u_k$.

b) There exist kernels $k\in \mathscr{C}(X\times X)$ and orthonormal families of proper functions $(f_n)$ of the endomorphism $u_k$ of $L^2_{\mathbf{C}}$(X), formed of bounded functions, such that the sequence $(\|f_n\|_{\infty})$ is not bounded.