I don't think that this is correct. Perhaps that what Georgi had in mind was this: with respect to the usual inner product, P and I-P are orthogonal. Therefore, if the representations was ...

I find your proof unclear. I would write it as follows: Claim: (M,D) is a discrete metric space. Proof: The conditions for D to be a metric are obviously satisfied. Let x\in M, then the open ...

For any x\in X what do the open balls B(x,r)=\{y\in X; d(x,y)<r\} for positive real r look like? What happens for r\lt1? How do they look like if r\ge1? You should see that if r<1, then B(x,r)=\{x\} ...

You are right; \bigl\{\{x\}\bigr\} is a a countable (finite, actually) system of neighborhoods of x , for each x\in X . Note that, with respect to the discrete topology, V is a ...

It represents the identity map. The notation also implies that you could think of the diagram as being the same as a triangle, with the two Xs being squished together (kind of like a quotient of the ...

a) In such exercises, you are allowed to use any metric and mapping you wish to. A nice one for counterexamples is d(x,y) := \begin{cases}0, &x = y\\1, & x\neq y\end{cases} What is the ball B_r(x)=\{y\in\mathbb R\mid d(x,y)<r\} ...