Any good math book reading suggestions?

Just started my freshman year at community college and I’m not taking any math classes this semester so I checked out a couple of math books from the library to read when I can. I’ve started with ” The Golden Ratio: The Story of Phi, the World’s Most Astonishing Number ” by Mario Livio and I’m loving it so far so I’m curious if anyone knows any other math books similar to that relating to calculus or trigonometry. I’m really not at the point where proof after proof can really keep my interest but I do enjoy the background of the discoveries and origins of the methods.

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What exactly is an Ito integral?

When constructing the Ito integral for simple processes, there is a natural intuition in terms of “betting strategies” that can only change at discrete times.

Now when constructing the Ito integral for a general continuous integrand, what we do is approximate the integrand pointwise in L2 by simple processes and then take the L2 limit of the corresponding integrals.

What does this represent? What exactly is an L2 limit of simple integrals? And why L2 in particular?

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Math Students: What do you take notes on?

I need some notebooks for the coming school year and I can’t find any online that I really like. I just need them for doing homework and taking notes from books and lectures. I also have pretty small handwriting and I find that most notebooks have lines that are an awkward size.

Any suggestions?

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Local homeomorphism lifting theorem

Hi, I’d greatly appreciate it if someone could illuminate the last three lines of this proof

Here is some context to aid in reading it:

[; C(I;Y) ;], where [; I ;] denotes the unit interval, is the space of all paths in [; Y ;], endowed with the topology generated by the following basis:

Let [; cup Ui ;], [; P = { t_0=0 ,t_1,…, t_n=1 } ;], be a collection of open sets where each [; U_i ;] comes from some basis for the topology on [; Y;] and [; P ;] is a partition on the unit interval [; I ;]. Furthermore the union is over the following set of indices: [; J = {1,2,…,n } ;]. (I wasn’t able to typeset a union no matter how hard I tried.. for whatever reason). Then the set [; A(t_0,t_1,..,t_n;U_1,U_2,..,U_n) ;] is the set of all paths [; a in C(I;Y) ;] such that [; a([t{i-1},t_i]) in U_i ;] for all [; i ;]. We endow [; C(I;X) ;] with the same topology.

[; f : X rightarrow Y ;] a local homeomorphism of topological spaces has the “unique path lifting property”, if for all paths [; a in C(I;Y) ;] and [; x in X ;] such that [; f(x) = a(0) ;], there is exactly one lifted path [; tilde{a} in C(I;X) ;] such that [; tilde{a}(0) = x ;], the lift with respect to $f$.

I understand all of the proof besides the third to last line, since I see that [; tilde{b}([t_0,t_1]) = phi_1(b([t_0,t_1]) ;] as both lifts agree at [;t = t_0 = 0;], by construction since [;tilde{b}(0) = x’ in U_1;], so by unique path lifting they agree on [;[t_0,t_1];]. The issue I have is why this also holds for [;i=2,3,…,n;]

Thanks for reading !

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