Homework Statement
This is not a homework question. I am solving this from the lecture notes that one of my friend's has got from last year.
If C(X) denotes a set of continuous bounded functions with domain X, then if X= [0,1] and fn(x) = x^n. Does the sequence of functions {fn} closed ...
Homework Statement
I need to understand as to why the following series fn(x) = x/(1+n*x^2) is point wise convergent (as mentioned in the book of Ross) and not [obviously] uniformly convergent.
Homework Equations
The relevant equation used is that lim (n-> infinity) sup|(fn(x) -...
Homework Statement
Prove that every sequence of bounded functions that is uniformly convergent is uniformly bounded.
Homework Equations
Let {fn} be the sequence of functions and it converges to f. Then for all n >= N, and all x, we have |fn -f| <= e (for all e >0). ---------- (1)...
Homework Statement
I am trying to see the geometric interpretation of the generalized MVT. It is not a homework problem, but would like to know how to interpret the equation
Homework Equations
[f(b)- f(a)]* g'(x) = [g(b)- g(a)]* f'(x)
The Attempt at a Solution
On...
Homework Statement
If X is bounded non empty subset in R (usual) and f:X->R is uniformly continuous
function. Prove that f is bounded.
Homework Equations
The Attempt at a Solution
Since X is bounded in R, it is a subset of cell. And all cells in R
are compact.All bounded sub...
Homework Statement
1) If f is a continuous mapping from a matric space X to metric space Y. A E is a subset of X.
The prove that f(closure(E)) subset of closure of f(E).
2) Give an example where f(closure (E)) is a proper subset of closure of f(E).
Homework Equations
The...
Homework Statement
I always get confused between countably many vs. uncountable. I suppose if one can index the points of a set , then it is countable.
1)So, anything that is finitie is countable. Anything that is infinite is also countable?
Then what is uncountable, something that...
Homework Statement
If X is a metric space such that every infinite subset has a limit point,
then prove that X is compact.
Homework Equations
Hint from Rudin: X is separable and has a countable base. So, it has
countable subcover {Gn} , n=1,2,3..... Now, assume that no finite sub...
Homework Statement
If X be a metric space in which every infinite subset has a limit point, then X is separable.
This is a question from Rudin but I am having some difficulty just understanding how to use the hint.
Homework Equations
The hint as in the book is .
Fix delta >0, and...
Homework Statement
I am having somewhat a difficult time just understanding a simple concept. I am trying to prove that every open subset G of a separable metric space X is the union of a sub collection {Vi} such that for all x belongs to G, x belongs to some Vi (subset of G).
I am...
1. Homework Statement
Every separable metric space has countable base, where base is collection of sets {Vi} such that for any x that belongs to an open set G (as subset of X), there is a Vi such that x belongs to Vi.
2. Homework Equations
Hint from the book of Rudin: Center the point...
Homework Statement
Every separable metric space has countable base, where base is collection of sets {Vi} such that for any x that belongs to an open set G (as subset of X), there is a Vi such that x belongs to Vi.
Homework Equations
Hint from the book of Rudin: Center the point in a...
Homework Statement
Suppose (X,d) is a metric space. For a point in X and a non empty set S (as a subset of X), define d(p,S) = inf({(d(p,x):x belongs to S}). Prove that the closure of S is equal to the set {p belongs to S : d(p,S) =0}
Homework Equations
Closure of S = S U S' , where S'...
Homework Statement
If an open cell is defined as (a1,b1) X (a2,b2) X .... (an,bn) in R^n and closed cell is defined as [a1,b1] X [a2,b2] X .... [an,bn], then every open set in R^n contains an open-n-cell and a closed-n-cell.
Homework Equations
Def: An open set is a set which has all...
Homework Statement
In general, in R^n, what is the best way to approach the problem - a given set is open?
The given set E is such that for all x,y that belong to the given set, d(x,y) < r.
Homework Equations
The Attempt at a Solution
let x be the center of the sphere and y be...
Homework Statement
An open-n-cell in R^n defined as (a1,b1) x (a2,b2) x ....... (an, bn). Prove that every open n-cell is open.
Homework Equations
The Attempt at a Solution
I was thinking of using induction. Clearly the base case n=1 is open as (a1,b1) is open in R1. It is a...
Homework Statement
If E is subset of R^2, then is every point of every closed set E, a limit point of E?
Homework Equations
The Attempt at a Solution
I think the answer is yes. Consider E = { (x,y) | x^2 + y^2 <= r^2} , where r is the radius.
Consider a point p that belongs...
Homework Statement
We know that (a,b) is open in R. But is it open R^n?
Homework Equations
The Attempt at a Solution
I don't think (a,b) is open in R^n even if it is open in R. Let's take for example n=2, then
E = {(x,y) | x^2+y^2 < r^2} , where r^2 = |b|^2 + Y^2 , for all...