PROBLEM SHEET 1.
Problem 0.1. Let Z be a normed space, let X be a finite-dimensional subspace of Z and let {x1 , x2 , . . . , xn } be a basis
of X. Show that there exist h1 , . . . , hn ∈ Z ∗ such that hi (xj ) = δi,j (Kronecker delta). [Note that this is not “just” linear
algebra; continuity matters too.]
Pn
By defining P (z) = j=1 hj (z)xj , deduce that X is complemented in Z.
Now let Y be any closed subspace of Z with X ∩ Y = {0}. Show that X + Y is a topological direct sum. [You may wish
to exploit Theorem 1.2 and the compactness of bounded closed sets in X]
Problem 0.2. Let X and Y be the subspaces of the Hilbert space `2 defined by
X = {(x1 , x2 , . . . ) ∈ `2 : x2n = 0 for all n ∈ N};
Y = {(x1 , x2 , . . . ) ∈ `2 : x2n−1 = nx2n for all n ∈ N}.
Show that X and Y are closed subspaces but that the sum X + Y is not closed and hence not complete. Deduce that
X + Y is not a topological direct sum. Letting P denote the linear projection on X + Y defined by P (x + y) = x when
x ∈ X, y ∈ Y , show directly that P is not continuous by exhibiting vectors zn ∈ X + Y with kzn k = 1 and kP (zn )k → ∞.
Problem 0.3. Let X be a separable Banach space that is isomorphic to the Hilbert space `2 .
(1) Show that every closed subspace of X is complemented in X. 1
(2) Show that every infinite-dimensional closed subspace of X is isomorphic to X.
(3) Show that every infinite-dimensional quotient space of X is isomorphic to X.
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Problem 0.4. The Lindenstrauss–Tzafriri theorem (see the preceding question) makes it clear that, on general Banach
spaces, complemented subspaces are the exception, rather than the rule. Here is a way to exhibit such subspaces in a
familiar space.
(1) Let the Banach space Z be the topological direct sum X ⊕ Y . Show that Y is isomorphic to the quotient space
Z/X.
(2) Show that every separable Banach space is isomorphic to a quotient of `1 .
(3) Assuming the result (to be proved in the next chapter) that every infinite-dimensional subspace of `1 has nonseparable dual, prove that `1 has (many) non-complemented subspaces.
Problem 0.5. Let Z be a subspace of a normed space X.
(1) Show that the bidual Z ∗∗ is identifiable with the subspace Z ⊥⊥ of X ∗∗ , with JZ being the restriction of JX to Z.
(2) Show that if X is reflexive and Z is a closed subspace then both Z and X/Z are reflexive. 3
1The Lindenstrauss–Tzafriri Theorem (1971) establishes the converse: if every closed subspace of a Banach space X is complemented then
X is isomorphic to a Hilbert space.
2The analogous converse here was established only in the late 1990’s (Gowers + Komorowski–Tomczak)
3Thus any Banach space with a subspace isomorphic to c or ` is non-reflexive. Until 1951 and the space J of R.C James, this was the
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only known obstruction to relexivity.
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