Recent content by xeno_gear

Hello, I'm reading through John Conway's A Course in Functional Analysis and I'm having trouble understanding example 1.5 on page 168 (2nd edition): Let (X, \Omega, \mu) and M_\phi : L^p(\mu) \to L^p(\mu) be as in Example III.2.2 (i.e., sigma-finite measure space and M_\phi f = \phi f is a...

A is arbitrary and isn't necessarily diagonalizable. There's an SVD and a Schur factorization though.

Homework Statement Let A \in \mathbb{C}^{n \times n} and set \rho = \max_{1 \le i \le n}|\lambda_i|, where \lambda_i \, (i = 1, 2, \dots, n) are the eigenvalues of A. Show that for any \varepsilon > 0 there exists a nonsingular X \in \mathbb{C}^{n \times n} such that \|X^{-1}AX\|_2 \le...

I've been following the thread.. seems interesting. Will the f(0) = 0 argument work? We only know that |f(z)| \le |z|^{1/2} on \mathbb{C} \backslash \{0\} .. or does the fact that f is holomorphic make it work?

OK, so I couldn't get it doing that either.. Had trouble with characteristic functions and boundedness of sets blahblahblah.. But I ended up trying this other thing in the analysis bag o' tricks that I forgot about.. I'll put it here in case anyone else has a similar question later and finds...

We aren't guaranteed that we know any specific method to get a y. The FTC just guarantees that an antiderivative exists. Existence means just that -- it exists, it's out there, but existence and construction are two different things. We're only guaranteed existence here but that doesn't...

oh yeah, "bootstrapping" as they call it in McDonald & Weiss. I tried that on the analogous problem (with the x^2 and 1/t^2) a week or so ago and couldn't get it to work.. but now I see why! Thanks for all your help!

Obviously I need to brush up on multiple integrals.. but anyway, I get this: \begin{align*} \int_0^t xf(x)\, dm(x) &= \int_0^t \int_0^x f(x) \, dm(u) \, dm(x)\\ &= \int_0^t \int_u^t f(x) \, dm(x) \, dm(u) \end{align*} which I'm pretty sure is...

argh, of course! I do that too often. Will try again, thanks!

I get: \begin{align*} \int_0^t xf(x)\, dm(x) &= \int_0^t f(x) \int_0^x 1 \, dm(u) \, dm(x)\\ &= \int_0^t \int_0^x f(x) \, dm(u) \, dm(x)\\ &= \int_0^x \int_0^t f(x) \, dm(x) \, dm(u)\\...

I haven't. This is a qualifying exam problem from 4 years ago so I'm not sure what their class did, although generally anything goes as long as we justify it. The formula is the same but we need to throw in hypotheses about integrability, right?

Homework Statement Suppose f \in L^1((0,\infty), dm). Prove that \[ L := \lim_{t\to\infty} \frac{1}{t}\int_0^t x\, f(x)\, dm(x) = 0. \] Homework Equations dm(x) represents integration with respect to Lebesgue measure. The Attempt at a Solution I know how to do this if f is...

Got it, thanks!

Homework Statement Here's an old qualifying exam problem I'm a little stumped on: Let (X,\mu) be a \sigma-finite measure space and suppose f is a \mu-measurable function on X. For t > 0, let \[ \phi(t) = \mu(\{x \in X : |f(x)| < t \}). \] Prove that \[ \int_0^{\infty}...

Hello. I'm studying for an ODE/PDE qualifier and I'm wondering how to do this problem. I feel like it should be pretty easy, but anyway.. Show that a Hamiltonian system in \mathbb{R}^{2n} has no asymptotically stable critical points. Any suggestions? Thanks..