FUNCTIONAL ANALYSIS LECTURE NOTES CHAPTER 3. BANACH ...

CHAPTER 3. BANACH SPACES 29
Exercise 4.23. Let M be a closed subspace of a Hilbert space H, and let π be the canonical
projection of H onto H/M. Prove that the restriction of π to M ⊥ is an isometric isomorphism
of M ⊥ onto H/M.
Remark 4.24. There is no analog of the preceding result for arbitrary Banach spaces. If X
is a Banach space and M is a closed subspace then we say that M is complemented in X if
there exists another closed subspace N such that M ∩ N = {0} and M + N = X.
It is not true that every closed subspace of every Banach space is complemented. In
particular, c0 is not complemented in ℓ ∞ . Also, if 1 < p ≤ ∞ and p = 2, then ℓ p has
uncomplemented subspaces.
Next we will define the product or direct sum of normed spaces. An analogous definition
holds for the case of a finite collection of spaces.
Definition 4.25. Let {Xi}i∈N be a countable family of Banach spaces, and let · i denote
the norm on Xi. Define
For 1 ≤ p < ∞, define
For p = ∞, define
p Xi =
∞
i=1
∞ Xi =
Xi =
f = (f1, f2, . . . ) : fi ∈ Xi .
f ∈
∞
Xk : fp =
i=1
f ∈
∞
i=1
∞
i=1
fi p
1/p
i < ∞ .
Xk : f∞ = sup fii < ∞
i
.
Exercise 4.26. Let {Xi}i∈N be a countable family of Banach spaces and fix 1 ≤ p ≤ ∞.
Let X =
p Xi. Prove the following.
(a) X is a normed space.
(b) For each i, the projection Pi : X → Xi given by Pi(f1, f2, . . . ) = fi is continuous, and
Pi = 1.
(c) X is a Banach space if and only if each Xi is a Banach space.
Exercise 4.27. Let X1, . . . , Xn be finitely many normed spaces. Prove that the spaces ⊕pXi
are equal for 1 ≤ p ≤ ∞, and that all the norms · p are equivalent. For this reason, we
often denote this space by X1 × · · · × Xn.