METRIC SPACES TEST QUESTIONS AND VERIFIED ANSWERS

METRIC SPACES TEST QUESTIONS AND
VERIFIED ANSWERS

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

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

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

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

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

Topological equivalence - ANSWER Two metrics d_1, d_2 are called
topologically equivalent if d_1 open and d_2 open sets collide.

Isometric - ANSWER The metric spaces (M_1, d_1) and (M_2, d_2) are
called isometric if there exists a function f:M_1→M_2 such that:
1. f is bijective
2. d_2(f(x),f(y)) = d_1(x,y) for all x,y in M_1
Such a function is called an isometry.

Homeomorphic - ANSWER Two metric spaces M_1, M_2 are called
homeomorphic if there exists an f:M_1→M_2 such that:
1. f is bijective
2. U is open in M_1 iff f(U) is open in M_2