Real II

Question 1

topology, open, closed, ex (metric space), indiscrete topology, discrete topology, relative topology

Answer 1

1

Question 2

closure, interior, boundary

Answer 2

2

Question 3

limit point, proposition (closure of A)

Answer 3

3

Question 4

local base, base, subbase, Theorem (TFAE, 3), Theorem (generates)

Answer 4

4

Question 5

product topology, proposition

Answer 5

5

Question 6

first countable, second countable, proposition

Answer 6

6

Question 7

convergence in topological space, proposition (point in closure IFF)

Answer 7

7

Question 8

cofinite topology, cocountable topology

Answer 8

8

Question 9

T1/T2/T3/T4

Answer 9

9

Question 10

continuous at x, Proposition (TFAE, 4)

Answer 10

10

Question 11

weak topology

Answer 11

11

Question 12

product space theorems (3 of em)

Answer 12

12

Question 13

C(X), C_b(X), Theorem

Answer 13

13

Question 14

Urysohn’s Lemma

Answer 14

14

Question 15

Tiktze Theorem (2 versions), completely regular

Answer 15

15

Question 16

directed set, net, tail

Answer 16

16

Question 17

(wrt nets) frequently in, eventually in, x_\alpha -> p, subnet, ex, Theorem 1 (TFAE, 2), Theorem 2, Theorem 3

Answer 17

17

Question 18

notions of compactness (1,1’,2,3,3’,4), 3 theorems

Answer 18

18

Question 19

universal net, Lemma, Theorem (net compactness; TFAE, 4)

Answer 19

19

Question 20

locally compact

Answer 20

20

Question 21

Tychonoff Theorem

Answer 21

21

Question 22

equicontinuous, pointwise bounded

Answer 22

22

Question 23

Arzela-Ascoli (3 versions)

Answer 23

23

Question 24

Stone-Weierstrass

Answer 24

24

Question 25

complete, Theorem

Answer 25

25

Question 26

Theorem (linear equivalence; TFAE, 4)

Answer 26

26

Question 27

invertibility (theorem)

Answer 27

27

Question 28

algebra quotient, norm

Answer 28

28

Question 29

Hahn-Banach

Answer 29

29

Question 30

Applications of Hahn-Banach (3 of em)

Answer 30

30

Question 31

reflexive, Theorem (4 things)

Answer 31

31

Question 32

Baire Category, first category, second category

Answer 32

32

Question 33

uniform boundedness principle

Answer 33

33

Question 34

Banach-Steinhaus

Answer 34

34

Question 35

little open mapping theorem, open mapping theorem

Answer 35

35

Question 36

closed graph theorem

Answer 36

36

Question 37

TVS, ex, locally convex, theorem

Answer 37

37

Question 38

gauge function, 3 properties

Answer 38

38

Question 39

separation theorem / geometric Hahn-Banach + 3 corollaries

Answer 39

39

Question 40

topologies on X*

Answer 40

40

Question 41

Banach-Alaoglu, Corollary

Answer 41

41

Question 42

Goldstine

Answer 42

42

Question 43

2 random theorems!!

Answer 43

43

Question 44

separates points, completely regular, Proposition

Answer 44

44

Question 45

L^p, norm, L^infty

Answer 45

45

Question 46

Riesz-Fisher

Answer 46

46

Question 47

Holder's inequality, equality when, ||f||_p = ??, Alternate Holder inequality

Answer 47

47

Question 48

Minkowski

Answer 48

48

Question 49

simple functions in L^p, continuation of Holder

Answer 49

49

Question 50

dual of L^p, Theorem, Corollary

Answer 50

50

Question 51

relations between L^p as p varies (3 theorems)

Answer 51

51

Question 52

Chebychev's inequality, Theorem

Answer 52

52

Question 53

Distribution functions, \lambda_f, proposition (4 properties), Theorem

Answer 53

53

Question 54

weak L^p

Answer 54

54

Question 55

compatible couple, ∑(X), \Delta(X)

Answer 55

55

Question 56

interpolation pair, exact, of exponent t

Answer 56

56

Question 57

\H(X), norm, X_t, Theorem, Theorem

Answer 57

57

Question 58

Riesz-Thorin Theorem

Answer 58

58

Question 59

sublinear, strong type (p,q), weak type (p,q), Marcinkiewicz Interpolation Theorem

Answer 59

59

Question 60

Baire sigma-algebra, Lemma, norm, mapping J

Answer 60

60

Question 61

Boolean Theorem, TFAE (5)

Answer 61

61

Question 62

(inner/outer) regular, Theorem, Corollary

Answer 62

62

Question 63

dual of C(X)

Answer 63

63

Question 64

extreme point, Krein-Milman Lemma, Krein-Milman Theorem, remarks (2)

Answer 64

64

Question 65

examples of extreme points

Answer 65

65

Question 66

extreme points of C(K): Proposition, Theorem, Proposition

Answer 66

66

Question 67

Banach-Stone

Answer 67

67

Question 68

Milman

Answer 68

68

Question 69

affine transformation, equicontinuous, fixed point, Kakatani fixed point theorem

Answer 69

69

Question 70

topological group, Theorem (Hair measure)

Answer 70

70

Question 71

convex hull for complex dudes: Theorem, Theorem

Answer 71

71