19.2.2.12 Convergence in normed spaces

condition: Investigate the sequence of operators \( \left\{A_{n}\right\} \subseteq L(X, X) \) for pointwise and uniform (in norm) convergence in the following cases: a) \( X=l_{2}, \quad A_{n} x=\left(x_{1}, \ldots, x_{n}, 0, \ldots\right), \quad x=\left(x_{1}, x_{2}, \ldots\right) \in l_{2} \). b) \( X=l_{2}, \quad A_{n} x=\left(x_{n+1}, x_{n+2}, 0, \ldots\right), \quad x=\left(x_{1}, x_{2}, \ldots\right) \in l_{2} \). c) \( X=C[0,1], \quad\left(A_{n} x\right)(t)=t^{n}(1-t) x(t), \quad t \in[0,1] \). e) \( X=C[0,1], \quad\left(A_{n} x\right)(t)=t^{n} x(t), \quad t \in[0,1] \).