let $ \varphi \in C [ 0 , 1 ] $, define:
$ M_{\varphi} : L^{2} [ 0 , 1] \longrightarrow L^{2} [ 0 , 1] \quad M_{\varphi} ( f ) = \varphi f $
Is the following statement true?
1:$ M_{\varphi} $ is bounded?
2:$ M_{\varphi} $ is normal?
3: $ M_{\varphi} \geq 0$ if only if$ \varphi > 0 $?
Hints. (1) To show that $M_\phi$ is bounded, just write down $\|M_\phi f\|_2$, and use (generalized) Hölder to estimate it with $\|\phi\|_\infty \|f\|_2$.
(2) To check, whether $M_\phi$ is normal, start with writing down $\langle M_\phi f,g\rangle$, transform it to $\langle f,M_\phi^* g\rangle$, read of $M_\phi^* = M_{\bar \phi}$. Now use commutativity of multiplication of functions to show normality.
(3) Obviously not. Consider $\phi = 0$.