Recent content by antiemptyv

Homework Statement (This is for a functional analysis course.) Let C(X,Y) be the space of continuous functions X \to Y. Let \mathcal{C}_0 = \{\varphi \in C([0,\infty),\mathbb{R}) : lim_{x\to \infty}\varphi(x) = 0 \}. Let \mathcal{C}_0^1 = \{\varphi \in \mathcal{C}_0 : \varphi' \in...

Homework Statement Let L be the Lie algebra sl(n, F) and X = (x_{ij}, Y = (y_{ij}) \in L. Prove \kappa(X,Y) = 2n Tr(XY), where \kappa(,) is the Killing form and Tr() is the trace form.Homework Equations For any unit matrix E_{ij} and any X \in L, XE_{ij} = \sum_{m=1}^n x_{mi} E_{mj} and...

Homework Statement I am to compute the homology groups H_*(\mathbb{C}P^2 \times \mathbb{R}P^2; \mathbb{Z}_2), with coefficients in \mathbb{Z}_2. Homework Equations Kunneth formula and universal Coefficient theorem The Attempt at a Solution First I need the homology groups...

i don't quite follow... is that function in the space?

Suppose that ||f||= int 01| f(x) | dx and f is a piecewise continuous linear function on the interval [0,1]. If ||| f ||| = int 01 x | f(x) | dx, determine if the two norms are equivalent. I know the first defines a norm, and the space is not complete. Can anyone offer any hints as to...

yeah, that seems good. There's a generalization of the binomial coefficient--the multinomial coefficient that makes counting things like this simpler. (the hard part is deciding if it's the right approach!) It takes into account the overcount you initially were considering by dividing by the...

Great. Ahh, yes, that would certainly need to be shown. Thanks. I just wanted to make sure I'm getting these basic ideas down correctly, and not missing something completely obvious. We're just beginning Galois theory, and I'm using a couple supplementary texts because the one we use in...

What if we simplified the problem? Suppose we were making a 2-topping pizza from 3 possible toppings (given the same conditions)? Say the toppings are 1, 2, and 3. We can list the possibilities pretty easily. By letting "0" be the null topping, we can have the following toppings choices: 00...

Homework Statement Is i \in \mathbb{Q}(\alpha), where \alpha^3 + \alpha + 1 = 0? Homework Equations The Attempt at a Solution Suppose i \in \mathbb{Q}(\alpha). Then the field \mathbb{Q}(i) generated by the elements of \mathbb{Q} and i is an intermediate field, i.e. \mathbb{Q}...

ohh, i think i see now. 1 < a/b and, since f is decreasing, f(\frac{a}{b}) = \frac{a^{1/n}-(a-b)^{1/n}}{b^{1/n}} < 1 = f(1) and the rest is just algebra to show a^{1/n} - b^{1/n} < (a-b)^{1/n}. look good?

Thanks for your quick reply. Let's see what we have here... f(1)=1 and f(\frac{a}{b}) = \frac{a^{1/n}-(a-b)^{1/n}}{b^{1/n}}.

Homework Statement Let a>b>0 and let n \in \mathbb{N} satisfy n \geq 2. Prove that a^{1/n} - b^{1/n} < (a-b)^{1/n}. [Hint: Show that f(x):= x^{1/n}-(x-1)^{1/n} is decreasing for x\geq 1, and evaluate f at 1 and a/b.] Homework Equations I assume, since this exercise is at the end of...

true. i just think it's a nice example of being able to play with a series to find an explicit formula, though this isn't the the most telling of its nature.

sort of a generating function approach. define a function f by the series: f(x) = 1 + x + x^2 + x^3 + ... now take a look at x*f(x): xf(x) = x + x^2 + x^3 + ... add them together: f(x) - xf(x) = (1 + x + x^2 + x^3 + ... ) - (x + x^2 + x^3 + ... ) = 1. notice the terms cancel out. so...

I misinterpreted the problem. i was thinking of G acting on H, as a subgroup, and not of G acting on H element-wise, by just permuting the elements around in H. this means if H was not in Z(G), then its orbit would have order 2,...,p-1, none of which divide |H| and |G| since p is prime and...