Types of Convergence + Boundedness

Question 1

Formal Power Series (centered at a)

Answer 1

• Let a be in R, c_0, …, be in R.

- The symbol sum(n=0,infinity)(c_n * (x-a)^n) is a formal power series centered at a.

Question 2

f(x) converges to L in Y as x converges to x_0 in E (where E is in X)

Answer 2

• lim(x->x_0,x in E) f(x) = L if:

for all e > 0, there exists delta > 0: d_Y(f(x),L) < e WHENEVER x IN E and d_X(x,x_0) < delta

Question 3

Pointwise Converge of a Sequence of Functions

Answer 3

• f_n converges pointwise to f on X if:
  for each x in X, f_n(x) -> f(x) in (Y,d_Y) as n -> infinity
• Equivalently:
  for all x in X, f(x) = lim(n->infinity) f_n(x)
• Idea: N depends on x and e.

Question 4

Pointwise Convergence and Limit Switching?

Answer 4

• Cannot necessarily switch limit evaluation with pointwise convergence.

Question 5

Uniform Convergence of a Sequence of Functions

Answer 5

• f_n -> f uniformly on X as n -> infinity if:
  for all e > 0, there exists N >= m : FOR ALL x IN X, d_Y(f_n(x),f(x)) < e whenever n >= N.
• Idea: if given e, figure out N that works for all n, meaning that N depends only on e.

Question 6

Bounded Function

Answer 6

• A function f:X->Y is bounded if f(X) is a bounded subset of Y, meaning:
  1. there exists a y_0 in Y, R>0 : f(X) is contained in B_Y(y_0, R)
  2. there exists a y_0 in Y, R>0 : d_Y(f(x),y_0) < R for all x in X

Question 7

Set of Bounded Functions

Answer 7

• Defined as B(X->Y)

Question 8

Sup Norm of the Set of Bounded Functions

Answer 8

• For f,g in B(X->Y), define d_infinity(f,g) = sup(x in X) d_Y(f(x),g(x))

Question 9

Set of Bounded, Continuous Functions

Answer 9

• Define as C(X->Y) contained in B(X->Y) where C contains all f in B where f is continuous.