September 3, 2011 Exercises–Set 1
1. Suppose T:X→Yis a linear map of a Banach space Xinto Banach
space Y. Let
A= inf{k:|T x | ≤ k|x| ∀ x∈X}
B= sup
x6=0 (|T x |
|x|)
C= sup
|x|≤1
{| T x |}
Show that A=B=C.
2. Suppose |·|1,|·|2are two norms on Rn. Prove that there are constants
C1>0, C2>0 such that for every x∈Rn,
C1|x|1≤ | x|2≤C2|x|1
3. Suppose T:X→Yis a one-to-one continuous onto linear map from
the Banach space Xto the Banach space Yand there is a constant
k > 0 such that |T x | ≥ kfor all |x|= 1. Prove that there is a unique
continuous linear map S:Y→Xsuch that S(Tx) = xfor all x. For
those who know some functional analysis: is the same conclusion true
for one-to-one continous onto linear maps without the assumption that
there is such a k?
4. Prove that every linear map from Rnto Rnis uniformly continuous.
5. Let I= [0,1] be the closed unit interval. Show that the closed unit
ball in the Banach space C(I, Rn) is not compact.
6. Show that the Schauder Fixed Point Theorem becomes false if either
of the compactness or convexity conditions does not hold.
7. A compact topological space Xhas the fixed point property or fpp if
every continuous self-map of Xhas a fixed point. Prove that this prop-
erty is preserved by homeomorphism. That is, if Yis homeomorphic
to Xand Xhas the fpp, then Yalso has the fpp.
8. Let f: [0,1] →[0,1] be continuous. Show that, given > 0, there is
a continuous g: [0,1] →[0,1] such that ghas only finitely many fixed
points and |f(x)−g(x)|< for all x∈[0,1].
