COURSE CHART
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Weeks
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Course Topics
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1
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The Real line and Euclidean space (ordered field, distance, Schwarz Inequality, The topology of Euclidean space |
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2
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Open sets, Interior of a set, closed sets
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3
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Accumulation points, Boundary and closure of a set |
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4
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Sequences in R^n, Convergent Sequences, Cauchy Sequence, Completeness, Infinite Series |
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5
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Compactness, the Heine-Borel theorem,
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6
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Connected sets,Nested set property, Path-connected sets |
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7
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Continuous Mappings ;continuity, Images of compact and connected sets |
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8
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Operations on continuous mappings, the boundedness of continuous functions on compact sets, |
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9
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Continuous Mappings ;Uniform continuity, Examples |
| 10 | Pointwise convergence, Uniform convergence |
| Midterm Exam | |
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11
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Uniform convergence of sequences of functions, T-test Uniform convergence of series of functions,
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12
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The Weierstrass M-test, Interchanging the places of limits and integrals of sequences of functions, |
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13
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Integration and differentiation of series, Power series, Interval of convergence, Taylor's expansion
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14
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Equicontinuity, Arzela - Ascoli Theorem, Examples, Review |