1. Show that the function
1
cannot be approximated by polynomials in the annulus
z
{z | r ≤ |z| ≤ R},
0 < r < R.
2. Show that if the weak topology on a normed space X is metrizable then X is finite dimensional.
Hint: show that if the weak topology is metrizable then there exists a sequence {fn }n ⊂ X ∗ such
that the sets {x | |fk (x)| < 1, k = 1, . . . , n}, n ∈ N, form a base of neighbourhoods of zero; conclude
that the Banach space X ∗ has at most countable Hamel basis.
3. Let H be a Hilbert space. Show that if H 3 ξn → ξ weakly and kξn k → kξk, then ξn → ξ in
norm.
4. Let X be one of the spaces c0 or `p , 1 < p < ∞. Show that a sequence {x(n) }∞
n=1 ⊂ X
(n)
(n)
converges weakly to zero if and only if supn kx k < ∞ and xk → 0 as n → ∞ for every k.
5. Let K be a compact space. Show that a sequence {fn }∞
n=1 ⊂ C(K) converges weakly to zero
if and only if supn kfn k < ∞ and fn (x) → 0 for every x ∈ K.
1