Analysis II
Year 2 · 7 sections · 82 concepts · 196 practice questions
What's covered
free1 · Uniform convergence11 concepts
- Overview 1 · The plan of this section
- Definition 1.1 · Pointwise convergence
- Example 1.2 · What pointwise convergence fails to preserve
- Definition 1.3 · Uniform convergence
- Proposition 1.4 · Sup-norm test for uniform convergence
- Proposition 1.5 · Cauchy criterion for uniform convergence
- Theorem 1.6 · Uniform limits of continuous functions are continuous
- Theorem 1.7 · Uniform convergence and integration
- Theorem 1.8 · Uniform convergence and differentiation
- Theorem 1.9 · Weierstrass M-test
- Check 1.10 · Check: proving or disproving uniform convergence
2 · Power series and special functions10 concepts
- Overview 2 · The plan of this section
- Theorem 2.1 · Power series converge uniformly inside the radius
- Corollary 2.2 · Termwise integration and differentiation
- Theorem 2.3 · Abel's theorem
- Example 2.4 · ln 2 and π/4 as sums
- Proposition 2.5 · Logarithm from the exponential
- Proposition 2.6 · Identity theorem for power series
- Example 2.7 · A smooth function that is not analytic
- Proposition 2.8 · When the Taylor series converges to the function
- Check 2.9 · Check: manipulating power series
3 · Improper integrals11 concepts
- Overview 3 · The plan of this section
- Definition 3.1 · Improper integrals
- Proposition 3.2 · Cauchy criterion for improper integrals
- Theorem 3.3 · Comparison tests
- Proposition 3.4 · Absolute convergence implies convergence
- Example 3.5 · The integral of sin x / x
- Theorem 3.6 · Integral test
- Definition 3.7 · The gamma function
- Proposition 3.8 · Properties of the gamma function
- Theorem 3.9 · Differentiation under the integral sign
- Check 3.10 · Check: does this improper integral converge?
4 · Metric spaces17 concepts
- Overview 4 · The plan of this section
- Definition 4.1 · Metric space
- Definition 4.2 · Normed space
- Example 4.3 · Standard metric spaces
- Definition 4.4 · Open balls and open sets
- Proposition 4.5 · Unions and intersections of open sets
- Definition 4.6 · Closed sets, interior and closure
- Proposition 4.7 · Describing the closure
- Definition 4.8 · Dense sets
- Definition 4.9 · Convergence in a metric space
- Proposition 4.10 · Closed sets contain their limits
- Definition 4.11 · Continuity between metric spaces
- Theorem 4.12 · Three descriptions of continuity
- Definition 4.13 · Equivalent norms
- Theorem 4.14 · All norms on a finite-dimensional space are equivalent
- Proposition 4.15 · Bounded linear maps and the operator norm
- Check 4.16 · Check: open, closed, continuous
5 · Completeness10 concepts
- Overview 5 · The plan of this section
- Definition 5.1 · Cauchy sequences and completeness
- Proposition 5.2 · Closed subsets of complete spaces
- Proposition 5.3 · ℝⁿ is complete
- Theorem 5.4 · C[a, b] with the sup norm is complete
- Example 5.5 · C[0, 1] with the L¹ metric is not complete
- Theorem 5.6 · Contraction mapping theorem
- Example 5.7 · Solving equations by contraction
- Theorem 5.8 · Picard's existence and uniqueness theorem
- Check 5.9 · Check: applying the contraction mapping theorem
6 · Compactness and connectedness14 concepts
- Overview 6 · The plan of this section
- Definition 6.1 · Sequential compactness
- Proposition 6.2 · Compact sets are closed and bounded
- Theorem 6.3 · Heine–Borel theorem
- Theorem 6.4 · Continuous images of compact sets are compact
- Corollary 6.5 · Extreme value theorem
- Theorem 6.6 · Continuous on compact implies uniformly continuous
- Definition 6.7 · Connected sets
- Definition 6.8 · Path-connected sets
- Proposition 6.9 · The connected subsets of ℝ are the intervals
- Theorem 6.10 · Continuous images of connected sets are connected
- Corollary 6.11 · Intermediate value theorem
- Proposition 6.12 · Path-connected implies connected
- Check 6.13 · Check: compactness and connectedness arguments
7 · Differentiation in ℝⁿ9 concepts
- Overview 7 · The plan of this section
- Definition 7.1 · The derivative as a linear map
- Proposition 7.2 · Uniqueness of the derivative and the Jacobian matrix
- Theorem 7.3 · Chain rule
- Theorem 7.4 · Continuous partial derivatives imply differentiability
- Theorem 7.5 · Mean value inequality
- Theorem 7.6 · Inverse function theorem
- Theorem 7.7 · Implicit function theorem
- Check 7.8 · Check: differentiability proofs
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Proposition 1.4 · Sup-norm test for uniform convergence
Compute : uniform iff it tends to . To disprove, chase a bad point .
Proof
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