Functional Analysis
Homework 1
Spring 2017
Due: Monday, March 6, 2017
1. If x, y, z are points in a metric space (X, d), show that
d(x, y) ≥ |d(x, z) − d(y, z)|.
2. Let X = {(x1 , x2 , · · · )|xi ∈ R,
d(x, y) =
∞
X
P∞
i=1 |xi |
p
< ∞} and
!1/p
|xi − yi |p
for x = (x1 , x2 , · · · ) and y = (y1 , y2 , · · · ).
i=1
Prove that (X, d) is a metric space if p ≥ 1.
3. Suppose that (X, dX ) and (Y, dY ) are metric spaces. Prove that the Cartesian product
Z = X × Y is a metric space with metric d defined by
d(z1 , z2 ) = dX (x1 , x2 ) + dY (y1 , y2 ),
where z1 = (x1 , y1 ) and z2 = (x2 , y2 ).
4. Let (X, dX ), (Y, dY ), and (Z, dZ ) be metric spaces and let f : X → Y , and g : Y → Z be
continuous functions. Show that the composition
h = g ◦ f : X → Z,
defined by h(x) = g(f (x)), is also continuous.
5. Prove that every compact subset of a metric space is closed and bounded. Prove that a
closed subset of a compact space is compact.
6. Let (X, d) be a complete metric space, and Y ⊂ X. Prove that (Y, d) is complete if and
only if Y is a closed subset of X.
7. Let (X, d) be a metric space, and let {xn } be a sequence in X. Prove that if {xn } has
a Cauchy subsequence, then, for any decreasing sequence of positive k → 0, there is a
subsequence {xnk } of {xn } such that
d(xnk , xnl ) ≤ k
for all k ≤ l.
8. Suppose that f : X → R is lower semicontinuous and M is a real number. Define fM :
X → R by
fM (x) = min(f (x), M ).
Prove that fM is lower semicontinous.
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9. Let f : X → R be a real-valued function on a set X. The epigraph epif of f is the subset
of X × R consisting of points that lie above the graph of f :
epif = {(x, t) ∈ X × R|t ≥ f (x)}.
Prove that a function is lower semicontinuous if and only if its epigraph is a closed set.
10. Suppose that {xn } is a sequence in a compact metric space with the property that every
convergent subsequence has the same limit x. Prove that xn → x as n → ∞.
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