Some Midterm Solutions
3.(iii) Let G by our open connected set.
Note that x y i there is a path : [0, 1] G with (0) = x, (1) = y denes an equivalence
relation.
Symmetry and reexivity are obvious.
Transitivity is proven just by concatenating paths.
We cl
Solution Set
Math 205a - Fall 2011
Problem Set 1
Problem 1 - 1 Show that f is well-dened, monotone, continuous and that it is constant on every
interval contained in the complement of the Cantor set.
It is easy to show that if f x has two expansions, an a
Math 205A Problem Set 5
Solutions
November 28, 2011
1
Problem 1
(i) For any closed ball B = B (y, ) that contains x the ball B (x, 2 ) contains B and
has volume 2n m(B ). Thus, for any x in the set cfw_x : M f (x) > we can nd a
1
closed ball B around it
Solution Set
Math 205a - Fall 2011
Problem Set 1
Problem 1 Show that if f is integrable on E then
|f (x + t) f (x)|dx = 0
lim
t0
E
First, notice that this problem only makes sense if we assume that f is integrable on E + (, ) or if we
extend f to be zero
Solution Set
Math 205a - Fall 2011
Problem Set 4
1
Problem 1 Let g Lq [0, 1] and dene a mapping G : Lp R by G(f ) = 0 f gdx. Show that G is a
bounded linear functional on Lp .
Linearity follows from the linearity of integrals. Boundedness follows from Hld
Solution Set
Math 205a - Fall 2011
Problem Set 1
Problem 1 Let be a non-negative continuous function on Rn such that
t (x) = tn (x/t). Show that if g C (Rn ) with compact support then
= 1. Given t > 0 dene
t (x)g (x)dx g (0)
t (g ) =
Rn
First notice that
Lecture notes for Math 205A
Lenya Ryzhik
December 4, 2008
Essentially nothing found here is original except for a few mistakes and misprints here and
there. These lecture notes are based on material from the following books: H. Royden Real
Analysis, L. Ev