1. Find a continuum of linearly independent vectors in the space c0 . Assuming that you
know the Zorn lemma conclude that each of the spaces c0 , c, `∞ , s has a basis of continuum
cardinality.
2. Let E0 be a subspace of a normed vector space E. Prove that
(i) p([x]) = inf y∈E0 kx − yk is a seminorm on E/E0 ;
(ii) the closure Ē0 of E0 is a vector space;
(iii)there is a canonical isometry between the normed space E/Ē0 and the normed space
obtained from (E/E0 , p).
3. Let E be a complex space with a scalar product. Let H be the completion of E with
respect to the norm kxk = (x, x)1/2 . Show that the scalar product on E extends to H, and
then H becomes a Hilbert space.
4. Let E be a normed real space. Prove that the norm is defined by a scalar product on
E if and only if it satisfies the parallelogram identity
kx + yk2 + kx − yk2 = 2(kxk2 + kyk2 ).
5. Let {ei }i∈I be an orthonormal system in a Hilbert space H. Prove that for any x ∈ H
the set
{i ∈ I | (x, ei ) 6= 0}
is at most countable.