Recent content by lunde

Which part isn't right?

I believe that it's actually the least upper bound property and not the nested interval property. In terms of your main question, the rational numbers form a countable dense subset of R, which allows us to use Riemann integrals on it. It is always possible to find a rational number (and thus...

Yeah, the idea of partitioning the basis makes total sense. For some reason I just hadn't been able to find anyone saying that this statement was true, so I started to get worried. But the reason I was interested in a Hilbert space being equal to the direct sum of 4 had to do with the compact...

Hello, I hope I am asking this in the right area of the forums. My teacher wrote the following formula down at our last meeting, and I was wondering if it was true ( \mathcal{H} is the infinite dimension separable Hilbert space): \mathcal{K} (\mathcal{H}) \approx \mathcal{K} (\mathcal{H}...

Hello, I hope I am asking this in the right area of the forums. I wanted to ask if the following formula is true (assuming H is infinite dimensional and separable): \mathcal{K} (\mathcal{H}) \approx \mathcal{K} (\mathcal{H} \oplus \mathcal{H} \oplus \mathcal{H} \oplus \mathcal{H})\approx...

How can you show that all surjective isometries between Hilbert spaces are linear?

Thanks. That's a cool way to show this, and then since it's an isometry it's bounded, great.

Hello, I had a question regarding unitary transformations. The most common definition I see for unitary transformations is defined as a transformation between Hilbert spaces that preserves inner products. I was wondering if all unitary transformations between Hilbert spaces (according to this...

Here's my best guess about this. The rationals are a countable dense set in \mathbb{R} . So give any function on \mathbb{R} you can approximate it arbitrarily close working strictly from a countable subset of your domain. This along with the fact that we can use a net to take the limit as...

Well order the real numbers, let {a_n}_{n \in S_{\Omega} } be the the singleton sets of odd numbers in the well order (i.e. skip a number, grab a number, skip a number, grab a number). Since the real numbers are T_1 then singleton sets are closed. \mathbb{R} as a topological space is...

Homework Statement I need to find the residue of e^(az)/(1+e^z)^2 at I Pi. For some reason this is such much harder than I thought it was going to be. Mathematica is not even helping :(. Homework Equations Cauchy's kth Integral formula. The Attempt at a Solution I made an...