19.2.2.10 Convergence in normed spaces

Condition: Show that the norms \( \|x\|_{c}=\max _{0 \leq t \leq 1}|x(t)| \) and \( \|x\|_{c_{1}}=\int_{0}^{t}|x(t)| d t \quad \) are not \( \quad \) equivalents to \( \quad \) in the space \( c[0,1] \). Hint: examine the sequence \( x_{n}(t)=t^{n} \) for convergence in these norms.