4.1.1 
Direct check

4.1.2
Direct check

4.1.3
Consider one norm on the unit sphere defined by the other norm.

4.1.4
Direct evaluation of the difference.

4.1.5
Direct check.

4.1.6
Express a convergent sequence as the sum of a convergent series of successive differences of the sequence.  Distribute the series members so that an absolutely convergent series is obtained.


4.4.5 
f=sum1 = f_n+fr_n, g=sum2=g_n+gr_n, absolutely and uniformly converging series (scalar
functions).
fr_n, gr_n  - the rests, n-> infty => fr_n, gr_n -> 0
Thus
fr_n(T), gr_n(T) -> 0 (in norm); f_n(T) -> f(T), g_n(T)-> g(T)

n,m -> infty f_n*g_m -> fg

f_n(T) g_m(T) = (f_n*g_m)(T) = (f-fr_n)(g-gr_m)(T) =
(fg)(T)  - fr_n(T)g(T) - ... =(fg)(T) +rest, rest -> 0 in norm

4.4.6 Take function sequence f_n,  whose norm equals 1 and whose support is between 1-1/n
and 1. n-> infty

5.1.5 as 4.4.6  The same idea. Take functions as in 4.4.6 but orthogonal to each other.
After multiplication by t they will remain "almost orthogonal", and of almost the same norm 1.
That means that there is no convergent subsequence.