Homework for Analysis 3
You can assume for simplicity that all the spaces are over R unless it is explicitly stated
that the space is over C.
Problem 1. Prove that `p , 1 ≤ p < ∞ is separable.
Problem 2. Prove that `∞ is not separable.
Problem 3. Prove that the spaces `13 and `∞
3 are not isometric.
Problem 4. Prove that the spaces c0 and c are not isometric.
Problem 5. Prove that if X is infinitely dimensional linear space, then there is a norm
k · k in X such that (X, k · k) is not complete.
Problem 6. Let x = (x1 , x2 , x3 , . . .) ∈ `∞ . Prove that the norm in the quotient space
`∞ /c0 is given by
k[x]k = lim sup |xn |.
n→∞
Problem 7. Let Y be a closed subspace of c consisting of constant sequences. Prove that
the quotient space c/Y is isomorphic to c0 .
Problem 8. Let M ⊂ C[0, 1] be a subset consisting of all functions f such that
Z 1/2
Z 1
f (t) dt −
f (t) dt = 1.
0
1/2
Prove that M is a closed and convex subset of C[0, 1] that has no element of minimal norm.
Problem 9. Let X be a Banach space and let Λ : X → `∞ be a linear operator, so that
Λ(x) = (Λ1 (x), Λ2 (x), . . .) is a bounded sequence of real numbers for every x ∈ X. Prove
that the operator Λ is bounded if and only if each linear functional Λn is bounded.
Problem 10. Prove that a two-linear operator B : X × Y → K is continuous if and only
if there is a constant C > 0 such that |B(x, y)| ≤ Ckxk kyk for all x ∈ X and y ∈ Y .
Problem 11. Let X be the vector space of all polynomials in one real variable, with norm
Z 1
kpk =
|p(t)| dt.
0
Consider the two-linear functional B : X × X → R defined by
Z 1
B(p, q) =
p(t)q(t) dt.
0
Show that B is continuous with respect to each variable p and q, but it is not continuous
on X × X.
Problem 12. Prove that every separable metric space X can be isometrically embedded
into `∞ . Hint: Consider a sequence (d(x, xk )−d(x0 , xk ))k , where x0 ∈ X and {xk }∞
k=1 ⊂ X
is a dense subset.
1
Problem 13. Prove that every separable real Banach space is isometrically isomorphic to
a subspace of `∞ .
Problem 14. Prove that there is an isometric embedding of R to real `∞
2 that is not
linear.
Problem 15. Let H be a real Hilbert space and let A : H → H be a bounded strictly
positive operator, i.e. there is β > 0 such that
hAx, xi ≥ βkxk2
for all x ∈ H.
−1
Prove that A is an isomorphism and kA k ≤ β −1 .
Problem 16. Let H be a Hilbert space over R and let B : H × H → R be a continuous
two-linear functional. In addition assume that B is strictly positive, i.e. there is β > 0
such that
B(x, x) ≥ βkxk2 for all x ∈ H,
Prove that for every y ∈ H there is a unique u ∈ H such that
B(u, x) = hy, xi for all x ∈ H.
Moreover kuk ≤ β
−1
kyk. Hint: Use Problem 15.
Problem 17. Let {un }∞
n=1 be an orthonormal basis in a Hilbert space H. Prove that if
∞
{vn }n=1 is an orthonormal set such that
∞
X
kun − vn k2 < ∞,
n=1
then
{vn }∞
n=1
is also an orthonormal basis.
2