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508Outline - Advanced Analysis Outline Math 508 Fall 2010...

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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 | : = 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. Euler’s 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 X is used to define a metric d ( X , Y ) : = X - Y . 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
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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 X 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 [ 0 , 1 ] with the uniform norm) that we will meet soon. In some normed linear spaces there is an inner product X , Y and the norm is given by X : = X , X . 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 X , Y = 0, in which case the Pythagorean theorem holds: X + Y 2 = X 2 + Y 2 .
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