The following homework is given to my daughter by her functional analysis teacher: Find the spectrum of the operator $Ax(t)=tx(t)$, $\,\,(0\leq t\leq 1)$, $$A:L^2[0,1]\rightarrow L^2[0,1].$$ Only thing I know is that the spectrum of a matrix is the set of the eigenvalues of that matrix. But, there is no matrix here. I tried to create a matrix by $A(t^k)=tt^k=t^{k+1}$ where $k\in\Bbb N$. That means, $A(e_k)=e_{k+1}$ where I thought of $L^2[0,1]=\Bbb R^{\infty}$. But, then it turns out that the matrix is $A$ is a lower triangular matrix with zero diagonal and hence its spectrum is $\{0\}$.
Is my answer correct? Or nonsense? Thanks in advance.