Chapter 7: Metric Spaces

Question 1

Metric space

Answer 1

Let X be a set and let D: X x X to Reals be a function such that (i) d(x,y) greater than or equal to 0 for all x, y in X

(ii) d(x,y) = 0 if and only if x=y
(iii) d(x,y) = d(y,x)
(iv) d(x,z) less than or equal to d(x,y) +d(y,z)

Question 2

Cauchy Schwarz Inequality

Answer 2

take x = (x1,x2,x3…xn) in R^n, and y = (y1,y2,…yn) in R^n then the sum of xjyj squared is less than or equal to the sum of xj squared times the sum of yj squared.

Question 3

restriction

Answer 3

Let (X,d) be a metric space and Y is a subset of X. Then the restriction d|yxy is a metric on Y.

Question 4

subspace

Answer 4

If (X, d) is a metric space, Y is a subset of X and d’ = d|yxy, then (Y, d’) is said to be a subspace of (X,d).

Question 5

bounded

Answer 5

Let (X, d) be a metric space. A subset S of X is said to be bounded if there exists a p in X and a B in R such that d(p,x) is less than or equal to B for all x in S. We say (X,d) is bounded if X itself is a bounded subset.

Question 6

Open Ball

Answer 6

let (X,d) be a metric space, x in X and delta >0. Then define the open ball or simply ball of radius delta around x as B(x, delta) = {y in X: d(x,y)<delta}

Question 7

Open

Answer 7

Let (X, d) be a metric space. A set V in X is open if for every x in V there exists a delta >0 such that B(x, delta) is in V.

Question 8

Closed

Answer 8

A set E in X is closed if the complement of E = X\ E is open.

Question 9

Open Neighborbood

Answer 9

If x is in V and V is open, V is an open neighborhood of x.

Question 10

Facts about Openness in Metric Spaces

Answer 10

1. the empty set and X are open in X
2. the finite intersection of open sets is open
3. the union of open sets is open

Question 11

Facts about Closedness in Metric Spaces

Answer 11

1. the empty set and X are closed in X
2. the intersection of closed sets is closed
3. the finite union of closed sets is closed