If you try to prove limit point compactness is equivalent to sequential compactness, it's actually rather natural. I sketched the proof below, so don't read it if you want to figure it out for yourself.
The main idea is that we can go back and forth between subsequences and infinite...
Well f is only defined if x is a natural number, so it's not well defined as a function from R to R. Also, we can't apply the summation rule if the number of terms is changing.
You should provide us some detail about what attempts you've made. From what you've given, I'm not sure what you're stuck on and how I can help. I'll assume you've tried the obvious thing, which is look at the definition, F is measurable if F^{-1}(a,\infty] is measurable. Since f is...
Well the identity function is a bounded linear map, so yes.
If you mean bounded linear functional, this is still true by Hahn-Banach, because we can send x to ||x||, and send each vector of the form ax to a||x||, where a is in C. By Hahn-Banach, this extends to a bounded linear functional.
I've studied some functional analysis at the level of Kolmogorov and Fomin, and that required a basic knowledge of linear algebra and real analysis. I also found this set of lectures online that might be useful if you're selfstudying or need another resource...