1. $\forall 1 \le p < q \le \infty \Rightarrow {\ell ^p} \subset {\ell ^q}$ (which means that ${\ell ^p}$ is in and, however, not equal to ${\ell ^q}$).
2. Two spaces $\left( {C\left[ {0,1} \right],{{\left\| \cdot \right\|}_\infty }} \right),\left( {{C^1}\left[ {0,1} \right],{{\left\| \cdot \right\|}_\infty }} \right)$, an operator $T:\mathop {C\left[ {0,1} \right] \to {C^1}\left[ {0,1} \right]}\limits_{f\left( x \right) \mapsto f'\left( x \right)} $, is $T$ bounded?
3. $T:\mathop {{L^2}\left[ {a,b} \right] \to {L^2}\left[ {a,b} \right]}\limits_{f\left( x \right) \mapsto \int_a^x {f\left( t \right)dt} } $, prove that $T$ is bounded.
4. $f \in {L^2}\left[ {0,1} \right]$ s.t. $\int_0^1 {{x^n}f\left( x \right)dx = \frac{1}{{n + 2}},\forall n \ge 0} $, prove that $f\left( x \right) = x$ a.e. on $\left[ {0,1} \right]$.
5. $X$ is a normed space over $F$, $f:X \to F$ is a linear functional. Then $f$ is continuous $ \Leftrightarrow $ $\ker f$ is closed in $X$.
6. $\dim X < \infty $
(a) ${x_n} \to x$ weakly $ \Rightarrow $ ${x_n} \to x$ in norm.
(b) ${T_n} \to T$ pointwise $\Rightarrow $ $T_n \to T$ in norm.