Lecture Notes

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====== Midterm 2 ======
In the second part of the course, we first covered basic notions of topology: open / closed / compact subset, in the framework of metric space (Rudin Ch2). Then, we studied the notion of continuous maps between metric spaces(Rudin Ch4). We give three characterizations of continuous maps, using $\epsilon-\delta$ language, using convergent sequences, and the most general notion: "preimage of open set is open". 
Continuous maps sends compact set to compact set, and sends connected set to connected set. We also discussed the notion of uniform continuity (property of a single function), and the notion of uniform convergence (property for a sequence of functions), not to be confused. 

The textbook to read is Rudin Ch 2, Ch 4, and Ch 7's first three sections. Ross's section 13-15, 17-22, 24, 25. We have mainly used Rudin's exercise for the homework, now for review purpose, you can take a look at Ross's exercises. 

Another source of practice problems is the book //A Problem Book in Real Analysis//, available to [[https://link.springer.com/book/10.1007/978-1-4419-1296-1 | download from springer]] (after you have authentication through Berkeley. You might have to allow for cross-site tracking in your browser for this to work.)

===== Logistics =====
The setup for exam is the same as last time. We will have usual exam time during class, 80 minutes, and you may scan your exam and upload to gradescope (note that this time, you need to specify the pages to the problems, as in your homework). The submission deadline is 10 minutes after the exam ends. 

For alternative time slot, we have 9pm and 10pm PT. Please let me know via zoom or email which one you need. This is only for students in a different time zone, more precisely, in the Asia or European time zones. 

For DSP students, you can submit directly to me via email or zoom. 

===== Summary of content =====
First, let's talk about topology of metric space. A set with the notion of distance is called a metric space. For a metric space, we define what it means to be an open set. Then, we verify that the collection of open sets satisfies the axioms of topology. We define closed set to be the complement of open sets. We introduced the notion of 'induced topology', and mentioned that openness and closedness are 'relative notion', i.e., we need to say $A$ is an open subset in which ambient space. 

The notion of compactness was introduced. The definition uses open cover. Then, we discussed the sequential compactness. One should know that for metric space, compactness is equivalent to sequential compactness (even though we did not give a proof). The various property of compact set: compact set is closed and bounded; closed subset of a compact set is compact. 

The notion of connected set is also introduced. A set is connected, if it cannot be written as a disjoint union of two non-empty open subset. 

In $\R^n$, we show that compactness is equivalent to closed and bounded. (Heine-Borel theorem). For $\R$, we showed that a connected set corresponds to an interval (open interval, closed interval, or half-open half-closed). 

Then, we defined what it means to be a continuous map. The three equivalent definitions: 1) pre-image of open is open 2) perserves convergent sequences 3) $\epsilon-\delta$ language. Then, we show that continuous map preserves compact set, preserves connected set. We also discussed discontinuities for function on $\R$. 

Then, we discussed about uniform continuity. This is a special notion for maps from a metric space to metric space. If says, $f: X \to Y$ is uniformly continuous, if for any $\epsilon >0$, we can find $\delta>0$, such that for any $x, y \in X$, if $d_X(x,y) < \delta$ then $d_Y(f(x), f(y))<\epsilon$. One should be able to give examples to tell the difference between uniform continuity and just continuity. A useful result is that, continuous function on a compact set is automatically uniform continuous. 

Finally, we talked about convergence of functions. There are several different notions: the weakest one is pointwise convergence, then we have uniform convergence. (There are other sorts of convergences, say $L^p$ convergence, but we do not discuss them here). We discussed many examples of how pointwise convergence can behave weirdly. 

Also, we talked about some left-over material in sequences and series, like how to test if a series is convergent or not. There is a ratio test, root test and for alternating series, a special test.
Lecture Notes

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