Recent content by happybear

why infinity is not a map into R?

Homework Statement If d= sup{|f(x)-g(x)|} where f, g are contuous function from R to R, why is d not a metric Homework Equations The Attempt at a Solution

so does that mean that no matter whether X is compact or not, there is no such a function?

sup{|f-g|} means that the maximum of the value. I try to find a function f approaching g, but this eem not to be the case

Homework Statement For all real number x, can you find a function f and g such that sup|f(x)-g(x)|=0 Homework Equations The Attempt at a Solution

Homework Statement According to Stone Weierstrass Theorem, there exists a polynomial such that lim P = |x|. Can anyone give me an example of P? Maybe within the range of 0.01? ie. |f-P|< 0.01 Homework Equations The Attempt at a Solution

Yes. That is what I meant. And what I want to show is that there are infinitely many functions of f. That is how I interpret the question, since they ask if the set is unbounded. Or do I misunderstand it

But there are no definition of f, that is the point. F will be any continuous function from [a,b] to R. Can I prove this set is infinite?

Homework Statement Suppose I have a set E that contains all continuous function f from [a,b] --> R, I think this is unbounded, but can I prove it? Homework Equations d(f1,f2)= sup{f1(x)-f2(x)} The Attempt at a Solution I want to show |d(f1,f2)|>M for some M, but I don't know if...

I understand this part. But this is follow the assumption that we know exp(A)=\sum1/n!(A)^n What if we don't know this theroem. All we know this is true for A is real number, not when A is matrix, then makes it hard to prove, right

Sorry about that. How can I show it is not onto, since I cannot find a B in Y, and calculate its A in X? Is there any other to prove it instead of using Jordan Normal Form

Homework Statement How do I prove a power series is onto? Since I cannot calculate directly, especially I haven't learned Jodarn Normal form. Homework Equations The Attempt at a Solution By showing 1-1, I tried ∑(1/n!)[(M)^n-(N)^n]=0, what can I conclude from this step?

Homework Statement If X = R with a special metirc, En = [n, infinity) () How can I show infinite intersection of En is empty? Homework Equations The Attempt at a Solution

I get it now.Thanx so much

How about the mapping f:N->{2,5}. Is this countable?