8.15: These should follow pretty immediately from the definition of
an adjoint.

1(a): Does not converge. (Every coordinate converges to zero, but
the sequence is not bounded.)

1(b): Converges weakly to zero. (The sequence is bounded, and every
coordinate converges individually to zero.)

1(c): Does not converge. (The sequence is bounded, but the first
coordinate does not converge).

2: The sequence does not converge in norm, but it does converge
weakly to zero. To prove that the sequence converges weakly, prove
that (i) it is bounded, and (ii) the inner product of phi_n with
any function exp(inx) is zero, and that all such functions form a
basis.

3: When A is self-adjoint, we know that e^(inA) is unitary, so
||e^(inA) u|| = ||u|| for all u. Thus the sequence (u_n) is bounded
and since the unit ball is weakly compact, it follows that
(u_n) must have a convergence subsequence.

4: This should follow pretty immediately from the definitions of
unitary and self-adjoint operators.

5: To prove that A is unbounded, test it against the Fourier basis
functions. To determine the infimum, switch to working in Fourier
space (using Parseval's formula). Express the effect of
A in Fourier space. You should find that A corresponds to
multiplication by 100 - 18 n^2 + n^4 (where n is the Fourier
index). Factor this to obtain 19 + (9 - n^2)^2 which is
strictly positive. The infimum is 19, obtained for instance
for u(x) = e^(3ix)/sqrt(2*pi) (since n=3 is the minimizer). The
equivalence is proved by proving that there exist finite c and C
such that
c (1 + n^4) <= 100 - 18 n^2 + n^4 <= C (1 + n^4).