Recent content by Edwin

I hope this is not overly speculative: I was wondering if seasonal temperature variations over large land masses can cause the ground to expand in such a way as to offset the straight line distance between two landmarks 730 kilometers apart by about 10 meters, or so? I had spoken a while back...

Does every vector space have a basis?

So linearity is the key to the proof? Or does a topological vector space being finite dimensional also play a role when it comes to being able to uniquely determine a linear mapping by how it maps basis elements? In other words, is the following statement true: Given a linear mapping L...

Good question: the following ideas may help. Let g: A -> B be a function from a set A into a set B. Definitions of one-to-one and onto: g is one-to-one iff for every a1, a2 contained in A, g(a1) = g(a2) implies that a1 = a2. g is onto iff for every b contained in B, there...

Hi Hurkyl, Thank you! That makes a lot more sense now. Best Regards, Edwin

Hi, I was playing around with Euler's Identity, and I found something (or at least I think I found something) interesting: It is a well known identity sin(z) = [exp(iz) - exp(-iz)]/(2*i), where z is any complex number, exp is the complex exponential function, and i is the imaginary...

Nevermind, HallsofIvy constructed one! Even better. Should the open cover be (1-1/n, 1/n) though?

Probably the easiest way to disprove this theorem, would be to find a counter example. The interval [0,1] is a topological space that has the Heine-Borrel property. It follows that every closed and bounded subset of [0,1], is compact. The interval (0,1) is an open subset of [0,1]...

I had a quick question on a part of a proof in chapter 1 of Functional Analysis, by Professor Rudin. Theorem 1.10 states "Suppose K and C are subsets of a topological vector space X. K is compact, and C is closed, and the intersection of K and C is the empty set. Then 0 has a...

Try the triq substitution of u = 3*sin(v) for sqrt[9 - u^2], here we just didn't factor out the -1.

That makes sense, then it would just be an integral of the form sqrt[a^2 - u^2], for a > 0.

Try rewriting the quadratic expression under the square root symbol in vertex form sqrt[a]*sqrt[(x-h)^2 + k^2)] , as follows: sqrt[-1]*sqrt[(x-5)^2 - 9], then let u = x - 5, and use the formula from a table of integrals for an integral of the form sqrt[u^2 - a^2], for a > 0, and then...

I am not sure I am interpreting the question correctly. But as far as I know, given any linear space (vector space) W, the trivial linear subspaces of W are the origin {0} in W, and W itself. Given any linear space X, it can be shown that the union of any of the following spaces {{0}...

Yyat wrote: If A is an open subset of a topological space X and is given the induced topology, then any subset B of A that is open in A is also open in X. (Try showing this!) I appreciate all of your help, and knowledge. I'll give it a try. Please let me know if I make a mistake in...

Thank you! Sorry about the inaccurate statements: when I went to post, I got the ideas mixed up in my mind. The induced topology looks like a real convenient way to prove openness: if you have a topological space X, and C is a subset of B is a subset of X, all you have to do to prove that...