Real Analysis

1. Define a "metric on a set"
2. What is the usual metric on R(or C)?
3. What is the usual metric on R^n?
4. What is the discrete metric on any set X?
5. Taxicab metric on R^n?
6. What is the p-norm metric on R^n?
7. How are the taxicab metric and p-norm related?
Bonus (already did it on first try): Prove that the max{|x_i - y_i|} is equivalent to the limit of the p-norm as p approaches infinity on x,y in R^n.
Note: We use the sup when n can be infinite.

Prove: d(x,y) <= d(x,z) + d(y,z) iff d(x,y) >=|d(x,z) - d(z,y)| for x,y,z in a metric space.

Prove: d(x,y) <= d(x,z) + d(z,y) iff d(x,y) <= d(x,z_1) + d(z1, z2) + ...+ d(zn, y) for x,y,z in a metric space.

Define a norm on a vector space V (over R or C)

1. What is the automatic metric that can be associated to any norm ||*|| on a vector space V?
2. Prove that this metric is in fact a metric over R^n.

1. Define a bounded subset of a metric space.
2. What is the set of bounded functions from a set S to R^k?
3. What is the uniform norm on the set of bounded functions (which is a vector space)?
4. Prove that the uniform norm is a norm (mostly prove the triangle inequality).

1. How can we think of R^n as a function?
2. How can we think of C^n of a function?
3. How do the above functions relate to the uniform norm?
4. What norm is this the limit of? Why are they different?

1. Define an inner product on a vector space V.
2. verify that the inner product on the complex numbers satisfies <z, ax + by> = conj(a)<z, x> + conj(b)<z,y>.
3. Verify that in any complex inner product space, <x+y,x+y> = <x,x> + 2*Re<x,y> + <y,y>
4. What is the standard inner product on R^n? On C^n?
5. Is the inner product linear in the second variable in R^n? In C^n?
6. How can we relate the usual norm on R^n to the inner product on R^n? C^n should be the same.

1. Define a subspace of a metric space.
2. Is the Torus/ circle a subspace of Euclidean space? Why?

1. Is X= {0,1}x{0,1}x... = {(x_1, x_2, ...) : x_i in {0,1} for all i} countable? If not, what is its cardnality.
2. What is the ultrametric inequality? Is it stronger than the triangle inequality or weaker?
3. define k(x,y) = min{i : x_i != y_i} (why is this possible?). Then define d(x,y) = 1/k(x,y) if x!= y and 0 if x=y. Prove the ultrametric inequality.

1. Define the open ball in a metric space.
2. Define the closed ball in a metric space.
3. What are the intervals in R for the closed ball and open ball centered at a with radius r?
4. Define a open set in a metric space.
5. Define a closed set in a metric space.

1. Prove open balls are open sets.
2. Prove closed balls are closed sets.

1. Prove that the union of any collection of open sets in a metric space is open.
2. Prove that any finite intersection of open sets is open (and why finite?).
3. Prove that any finite union of closed sets is closed.
4. Prove that any intersection of a collection of closed sets is closed.

1. Prove X and 0 are both open and closed.
2. If X=R^n, are there any other sets that are both open and closed?
3. Prove that a singleton set in a metric space is closed.
4. Prove that a finite subset of a metric space is closed.
5. Prove that (in R^n with the usual metric) open half-spaces ({x : x_i > c}, {x : x_i < c}) are open.
6. Similarly, show that closed half-spaces are closed.

1. Define a open box in R^n.
2. Define a closed box in R^n.
3. Why are they open/closed respectively?
4. Are most(?) sets open, closed, open and closed, or neither?

1. Give an example of a subset of R that is neither open nor closed (with proof).

1. Define the interior of a subset of a metric space.
2. Define the closure of a subset of a metric space.

1. How is the closure of a subset E of a metric space X related to whether or not E is closed?
2. How is the interior of a subset E of a metric space X related where or not E is open?

Remark 2.12. We observe that
i) int(E) is an open set, and is the largest open set contained in E.
ii) E^{bar} is a closed set, and is the smallest closed set containing E.
iii) E^{bar} = (int(E^c))^c and int(E) = ((E^c)^{bar})^c
Prove iii).

1. Prove x in int(E) iff there exists r > 0 s.t. B_r(x) \subsetOf E.
2. x in E^{bar} iff for every r > 0 B_r(x) \intersect E != 0.

In R,
1. What is int((a,b])?
2. What is int(Z)?
3. What is int(Q)?
4. What is (0,1]^{bar}?
5. What is Z^{bar}?
6. What is Q^{bar}?

1. Define the neighborhood of a point in a metric space.
2. Give an example of a set that is the neighborhood of one point but not another point and explain why.

Define the support of a function f: X -> R^k where X is a metric space

Define when a subset of a metric space is dense.

1. When is a metric space separable?
2. Is R separable? Why?
3. Does the fact that the R\Q (irrationals) are dense in R imply R is separable? Why?

1. Define an accumulation point (limit point/ cluster point) of a subset of a metric space.
2. What denotes the set of all accumulation points for a subset E of a metric space?

Assume X =R:
1. What is {1, 1/2, 1/3,...}' ?
2. What is {0, 1, 1/2, 1/3, ...}' ?
3. What is Z' ?
4. What is Q' ?
Bonus: Prove the above equalities.

1. Prove lemma: a in E' iff for all r>0, (E\{a}) \intersect B_r(a) !=0.
2. Prove E^{bar} = E U E'.

Define a bounded set in a metric space

CANTOR SET

1. Define a sequence in any set X