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 [{, 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 $. 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.