Metric Spaces UPDATED Exam Questions and CORRECT Answers

Metric Spaces UPDATED Exam Questions
and CORRECT Answers
Metric - CORRECT ANSWER - A metric d on a set M is a function d: MxM→R
satisfying (for all x,y,z in M):
1. d(x,y) is positive
2. d(x,y) = 0 implies x = y
3. d(x,y) = d(y,x)
4. d(x,z) is less than or equal to d(x,y) + d(y,z)


Open Ball - CORRECT ANSWER - Centered at a, with radius r:
B(a,r) = {x in M: d(x,a)<r}


Closed Ball - CORRECT ANSWER - Centered at a, with radius r:
¬B(a,r) = {x in M: d(x,a)<r or d(x,a)=r}


Bounded - CORRECT ANSWER - A subset S of M is bounded if there exists an a in M
and a positive r such that S is contained in B(a,r)


Norm - CORRECT ANSWER - A norm on a vector space V is a function ║·║:V→R such
that for all x,y in V:
1. ║x║ is positive
2. ║x║ = 0 implies x = 0
3. ║cx║ = |c|║r║ for all c in R
4. ║x+y║ is less than or equal to ║x║+║y║


Convex - CORRECT ANSWER - A ball is called convex if:
║x║,║y║ are less than or equal to one implies that for every positive a and b which add to 1,
║ax + by║ is less than or equal to 1

, Subspace (metric space) - CORRECT ANSWER - Let M be a metric space and H is a
subset of M. (H, d_H) is a subspace of M where d_H(x,y) = d_M(x,y)


Open - CORRECT ANSWER - A subset U of M is open IN M if for all x in U, there exists
a positive z such that B(x,z) is a subset of U.


Closed - CORRECT ANSWER - A subset U of M is closed IN M if its complement, M\U,
is open (in M).


Convergence - CORRECT ANSWER - A sequence x_k in M is convergent to x in M if
d(x_k,x) → 0.


Continuous - CORRECT ANSWER - Let (M_1,d_1), (M_2,d_2) be metric spaces and
f:M_1→M_2. f is continuous at a if:
For every y > 0, there exists a z > 0 such that for every x in M_1, then d_1(x,a)<z implies
d_2(f(x),f(a))<y.
f is continuous if it is continuous at every a in M_1.


Lipshitz Continuous - CORRECT ANSWER - Let (M_1,d_1), (M_2,d_2) be metric spaces
and f:M_1→M_2. f is Lipshitz continuous if there exists a real c such that:
d_2(f(x),f(y)) is less than or equal to cd_1(x,y) for all x,y in M


Distance from a set - CORRECT ANSWER - Let A be a non-empty subset of a metric
space M. Then the distance of x in M from A is:
d(x,A) := inf(d(x,z)) for all z in A.


Continuous (open sets relation for metric spaces) - CORRECT ANSWER - f:M_1→M_2
is continuous iff for every open subset U of M_2, ¬f(U) is open in M_1.