Part A Metric Spaces and Complex Analysis

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  1. Definition 1.3. (v^k)k∈N converges to w ∈ R^n if
  2. Definition 1.3.1: If f : R^n → R and a ∈ R^n then we say that f is continuous at a
  3. Define distance function
  4. define metric space
    • a pair (X, d) of a set and a distance function d: X × X → R≥0 satisfying the axioms for a distance function. If the distance function is clear from context, we may, for convenience,
    • simply write X rather than (X, d).
  5. Definition 2.4. Let (X, dX) and (Y, dY ) be metric spaces. A function f : X → Y
    is said to be continuous at a ∈ X if
    • for any e > 0 there is a δ > 0 such that for any x ∈ X with dX(a, x) < δ we have dY (f(x), f(a)) < e. We say f is continuous if it
    • is continuous at every a ∈ X.
  6. If (xn)n≥1 is a sequence in X, and a ∈ X, then we say (xn)n≥1 converges to a
    for any e > 0 there is an N ∈ N such that for all n ≥ N we have dX(xn, a) < e.
  7. A function f : X → Y is said to be uniformly continuous if
    • for any e > 0, there exists a δ > 0 such that for all x1, x2 ∈ X with dX(x1, x2) < δ
    • we have dY (f(x1), f(x2)) < e.
  8. give three common metrics on R^n
  9. define a norm
  10. what is 
  11. define an open ball and a closed ball
  12. define open, a neighbourhood and a topology
  13. define an interior
  14. define a topology, topological space and continuity of a function between two topological spaces
  15. define closed
  16. define a limit point and an isolated point
  17. define closure
  18. define the boundary of a subset of a metric space
  19. define an isometry
  20. define homeomorphism, homeomorphic
  21. what is a Cauchy sequence
  22. define a complete metric space
  23. weierstrauss M-test
  24. define Lipschitz map, contraction
  25. CMT
  26. define disconnected, connected
  27. define the connected component
  28. state the IVT

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Part A Metric Spaces and Complex Analysis
2017-10-23 08:16:57

part a metric spaces and complex analysis
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