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Unformatted text preview: Advanced Analysis: Outline Math 508, Fall 2010 Jerry L. Kazdan This outline of the course is to help you step back and get a larger view of what we have done. Since this is only an outline, I will often not explicitly state the precise assumptions needed for the assertion to hold. Important Sets R : T HE R EAL N UMBERS Primary features: they form a field , are ordered , and are complete . The completeness is the primary feature that distin- guishes from the much smaller field Q of rational numbers. The function | x | is a norm that measures the size of a real number. Basic concepts: convergence of a sequence, limit point of a sequence and of a set, open , closed , and bounded sets, Cauchy sequences , count- able and uncountable sets. Two important theores are the Bolzano- Weierstrass Theorem (bounded sequences have convergent subsequences) and the Heine-Borel Theorem (when open covers have a finite sub-cover). These theorems are associated with the compactness of a set. C : C OMPLEX NUMBERS : z = x + iy They also form a complete field but are not ordered. The norm | z | : = p x 2 + y 2 measures the size. The trian- gle inequality | z + w | | z | + | w | is basic. The concepts of convergence and limit point etc. extend immediately. The convergence of an infinite series arises frequently. We define e z , cos z , and sin z using power series. Eulers beautiful observation e ix = cos x + i sin x is valuable. G ENERALIZATION : R k , C k , ` 1 , ` 2 , the set of n k real or complex matrices. These are important normed linear spaces . The norm k X k is used to define a metric d ( X , Y ) : = k X- Y k . The concepts of convergence and limit points generalize immediately, as do those of open and closed sets, Cauchy sequence, compactness, etc. If A is a square matrix we define e A using the power series. 1 The concept of connectedness arises and is important. These spaces are all complete (proving the completeness of ` 1 and ` 2 takes some work). Subsets of these, such as the closed unit ball of points X where k X k 1 are also metric spaces, although they are not linear spaces (if X and Y are in the unit ball, X + Y might not be). Most of our metric spaces will simply be subsets of normed linear spaces. The point of introducing the examples ` 1 and ` 2 is that while closed bounded sets are compact in R k and C k , they are usually not compact in ` 1 or ` 2 or in other important examples (such as the set of continuous functions on [ , 1 ] with the uniform norm) that we will meet soon. In some normed linear spaces there is an inner product h X , Y i and the norm is given by k X k : = p h X , X i . These spaces are particularly easy to use and arise frequently in applications. In them one can define two vectors X and Y to be orthogonal if h X , Y i = 0, in which case the Pythagorean theorem holds: k X + Y k 2 = k X k 2 + k Y k 2 ....
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This note was uploaded on 03/06/2012 for the course MATH 508 taught by Professor Staff during the Fall '10 term at UPenn.
- Fall '10