University of Bergen
General Functional Analysis Problems 3
1) A function f : (M, d) → (N, ρ) is called the Lipschitz function if there is a constant K such that ρ(f(x), f(y))≤Kd(x, y). Show that any Lipschitz function is uniformly continuous. In particular, any isometry is uniformly continuous.
2) Is the sum of two uniformly continuous functions is a continuous function? Why? Is this true for the product of functions?
3)Suppose thatf :R→Ris a continuous function such that lim
x→±∞f(x) = 0. Prove thatf is uniformly continuous.
4) Let (X, ρ) and (Y, σ) be metric spaces with Y complete. Letf be a uniformly continuous mapping from a subsetE ⊂X intoY. Then there is a unique continuous extension g off from E toE; that is a unique continuous mapping g : E → Y such that g(x) = f(x) for x∈ E. Moreover, g is uniformly continuous.
5) Prove that M is compact if and only if every decreasing sequence of nonempty closed sets has a nonempty intersection; that is, if and only if, wheneverF1⊃F2⊃ · · · is a sequence of nonempty closed sets inM, we haveT∞
n=1Fn6=∅.
6)LetX be a metric space and letEbe its subset. Show that the following definitions of nowhere dense set E inX are equivalent. A setE is nowhere dense inX if
1. X\E is dense inX,
2. E contains no nonempty open set,
3. any ball B⊂X contains another ballB′, such thatB′∩E=∅.
7) Show that every finite subset ofRis nowhere dense. Is every countable subset of Rnowhere dense?
Show that the Cantor set is nowhere dense.
8)Prove that a subset of a complete metric space is residual if and only if it contains a denseGδ. Hence a subset of a complete metric space is of first category if and only if it is contained in an Fσ whose complement is dense.
9)Prove that ifE is of first category andA⊂E, thenAis also of first category. Show that if{En} is a sequence of sets of first category, then S
En is also of first category.
10) Is the set of rational numbers in [0,1] a Gδ set? Is there a real valued function on [0,1] which is continuous on the rationals and discontinuous in the irrationals?
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