Recent content by dmuthuk

Hi, I've been trying to prove this statement for a while now but haven't made much progress: Suppose f:[a,b]\to\mathbb{R} has the property that for each y in the image of f, there are EXACTLY two distinct points x_1,x_2\in [a,b] that map to it. Then, f is not continuous. Well, I...

Hi, I'm just asking this out of curiousity, but I have never really understood the difference between calculus and analysis. The only thing I can say is that calculus is basically applied analysis or analysis is the rigorous theory behind calculus. So, if the difference is just a matter of...

Thanks. That clears it up for me. So, I guess specifying a finite basis without an ordering is not very useful. I mean, I don't even see a way to talk about any finite set without first indexing the elements and assuming the natural ordering.

Hi, I was just wondering if there is something more to the concept of an ordered basis other than the fact that it is simply a basis which is ordered. The reason I'm asking this is because I don't know why some linear algebra books consider this important enough to make the distinction. I...

Thanks guys! I'll browse through these books and see which one will work out for me.

Hi, I'm a grad student in pure math and I'm trying to re-learn my linear algebra from scratch because I never learned it properly while I was an undergrad. Actually, the course I took in second year was aimed mostly at science students and so we never went into much depth (half the course was...

Yes, I believe so.

Given two left adjoints F,H:\mathcal{C}\to\mathcal{D} of a functor G:\mathcal{D}\to\mathcal{C}, how do we show that F and H are naturally isomorphic? This is my idea so far (I am working with the Hom-set defenition of adjunction): We need to construct a natural isomorphism \alpha. So, for...

Well, the reason I ask this question is because we sometimes treat sequences as functions from N into a set when it is intuitively obvious what we are talking about. Is this unecessary precision or are there situations where intuition can be misleading? For example, in Dummit and Foote, the...

Hi, we often come across certain constructions in algebra that make use of some "formal" sum or "formal" linear combination or "formal" string of elements. Because this term is never defined, I have always been a little uncomfortable when it comes up. For a specific example, consider the...

Hi, I am trying to sort out a few things about the dual and double dual of a normed space which has got me a little confused. Given a normed space X over \mathbb{R}, if Y is a subspace of X, what is the relationship between Y^* and X^*? Can Y^* be identified with some subspace of X^*...

Yes, I believe I am. So, I guess what I wanted to know is if there exists a proof that the reals can be well-ordered without AC.

We all know that the axiom of choice is equivalent to the existence of a well-ordering for any set. And, this of course implies that \mathbb{R} can be well-ordered, in particular. However, how do we know that the axiom of choice is actually needed in the case of the reals? That is, if we...

I totally agree with the argument of course. The thing is I have just been learning the basic concepts of set theory (Halmos' book) for the first time and so I guess I am a little pedantic about the rigour of such proofs. For instance, I have always thought of a function as just a function or...

You are arbitrarily choosing elements from an infinite collection of sets, so how can you do it without the axiom of choice??