MAT 360 Course Notes
1. Super-short history of Mathematical Analysis
2. Crash-course on elementary Logic: Statements
3. Negation (not), Conjunction (and), Disjunction (or),
Conditional (if ... then ...), Bi-conditional (... if and only if ...)
, Equivalent statements
4. Truth tables
5. DeMorgan's laws, Contrapositive, examples of equivalent statements
6. Open statements, universal and existential quantifiers
7. Negation of quantified statements
8. Some Set Theory: The concept of a set
9. Membership, subsets, equal sets, the empty set
10. Special sets in analysis: R (real numbers), Q
(rational numbers), Z (integers), N (natural numbers),
intervals on the real line
11. Union and intersection of two sets, disjoint sets
12. Properties of union and intersection; especially the fact that
they are associative and distributive
13. Difference sets, complement of a set with respect to a
bigger set.
14. Union and intersection of an arbitrary family of sets
15. DeMorgan's Laws for complements of unions and intersections
16. Cartesian product of two sets
17. Functions and their graphs
18. Image and inverse image of a set under a function
19. Restriction and Extension of a function
20. Injective (one-to-one), surjective (onto), and bijective functions
21. Countable sets
22. Intuitive discussion of number systems
23. Basic facts about Q; Intuitive notion of denseness
24. Ordered fields and examples
25. Intuitive discussion of completeness of R
26. Least upper bound (supremum) of a set in R
27. Properties of supremum
28. The completeness axiom for the real line R
29. Greatest lower bound (infimum) of a set in R and its
similar properties
30. Absoute value and its properties, viewing |x-y| as a
distance
31. The principle of mathematical induction and applications in
proving inequalities.
32. The concept of a sequence of points in a set
33. Sequences of real numbers, monotone sequences
34. The naive and formal definition of limit for sequences
35. Some worked out examples of the epsilon-N definition
36. The limit of a sequence (if any) is unique
37. Convergent sequences are bounded
38. Sandwich Lemma and applications in finding limits
39. Algebraic operations respect limits
40. Every increasing (resp. decreasing) sequence in R
which is bounded above (resp. below) converges.
41. Application of the above theorem in finding limits
42. The concept of a Cauchy sequence: convergent sequences are
Cauchy. In R the converse is also true. To prove this, we need
the following array of facts:
43. The concept of a subsequence
44. Every bounded sequence in R has a convergent subsequence
45. Every Cauchy sequence is bounded
46. If a subsequence of a Cauchy sequence converges to some
L, then the entire sequence converges to L.
47. Corollary (of the above 3 facts): Every Cauchy sequence in
R coverges.
48. The concept of a cluster point for a sequence; examples
49. x is a cluster point of a sequence iff there exists
a subsequence converging to x
50. A sequence converges to L iff every subsequence
converges to the same limit L
51. Corollary (of the above 2 facts): A sequence converges iff
it is bounded and has a unique cluster point
52. The concept of limsup and liminf for a sequence in
R; examples
53. Basic properties of limsup and liminf; in particular, the
fact that a sequence converges iff its limsup and liminf coincide.
54. The n-dimensional Euclidean space
55. Review of the concept of a vector space
56. Examples of vector spaces: Rn, the space of real
sequences which converge to 0, the space of real-valued function
defined on on interval [a,b]
57. Standard norm on Rn and its properties
58. The general notion of a norm on a vector space
59. Examples of norms on Rn which are not standard
60. Standard inner product on Rn and its properties
61. The general notion of an inner product on a vector space
62. Cauchy-Schwarz inequality
63. Applications of Cauchy-Schwarz inequality
64. Every inner product gives rise to a norm (but not vice versa)
65. The general notion of a metric space
66. Every norm gives rise to a metric (but not vice versa)
67. Examples of metric spaces: normed vector spaces (Rn,
C[a,b], etc.), discrete metric on a set, word metric on the finite set of
all binary words of length n
68. Balls in metric spaces
69. Open sets in metric spaces; examples
70. Balls are open; arbitrary unions and finite intersections of
open sets are open
71. Interior of a set; examples; interior of a set is open
72. Closed sets in metric spaces; examples
73. Finite sets are closed; arbitrary intersections and finite
unions of closed sets are closed
74. Accumulation points of a set; examples
75. Closure of a set; examples; closure of a set is closed;
a point x is in the closure of a set A iff every
neighborhood of x intersects A
76. Boundary of a set; examples
77. Closure = Interior union Boundary
78. Sequences in metric spaces; convergence of sequences;
Cauchy sequences
79. A point x belongs to the closure of a set A
iff there exists a sequence of points in A which converges to
x.
80. Basic facts for real sequences hold in metric spaces:
limits are unique; a sequence converges to L iff every
subsequence of it converges to L; convergent sequences are
Cauchy, but not vice versa in general
81. A sequence in Rn converges to L iff
the sequence of i-th components converges to the i-th
component of L for every i
82. Complete metric spaces; examples
83. Every Euclidean space Rn (with the standard
metric) is complete.
84. Closed subsets of complete metric spaces are complete.
85. Intuitive definition of compactness through examples
86. The concept of sequential compactness in a metric spaces
87. Sequentially compact sets are bounded and closed
88. Open covers and formal definition of compactness
89. Compact sets are bounded and closed
90. A set in a metric space is compact iff it is
sequentially compact
91. A subset of a Euclidean space Rn is compact
iff it is bounded and closed
92. The above result is not true in arbitrary metric spaces
93. Nested set property of compact sets in metric spaces; examples
94. Nested set property is not true if the sets are not
compact; examples
95. Examples of compact sets in fractal geometry generated by
the nested set property: Sierpinski's carpet, Cantor set
96. More examples of "wild" compact sets in the plane: von Koch
snowflake, Julia sets of polynomials
97. Continuous paths in a metric space
98. Path-connected sets; examples
99. Every interval in R is path-connected
100. Connectedness; examples; connectedness is really different
from path-connctedness
101. Every path-connected set is connected
102. A subset of the real line is connected iff it is an
interval
103. The intuitive notion of continuity
104. Definiton of continuity for maps between metric spaces
105. Alternative definitions of continuity: A map is continuous
iff it respects limits of convergent sequences iff the preimage of
every open set under the map is open
106. Elementary properties of continuous functions; algebraic
operations on real-valued continuous functions
107. Continuous images of (path-)connected sets are (path-)connected
108. Continuous images of compact sets are compact
109. The Extreme Value Theorem
110. The Intermediate Value Theorem
111. Applications of the Intermediate Value Theorem; The
"cheesecake dilemma"
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Math 360