Exam (elaborations)
METRIC SPACES TEST QUESTIONS AND VERIFIED ANSWERS
- Course
- Institution
METRIC SPACES TEST QUESTIONS AND VERIFIED ANSWERS....
[Show more]
Content preview
METRIC SPACES TEST QUESTIONS ANDVERIFIED 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
The benefits of buying summaries with Stuvia:
Guaranteed quality through customer reviews
Stuvia customers have reviewed more than 700,000 summaries. This how you know that you are buying the best documents.
Quick and easy check-out
You can quickly pay through credit card or Stuvia-credit for the summaries. There is no membership needed.
Focus on what matters
Your fellow students write the study notes themselves, which is why the documents are always reliable and up-to-date. This ensures you quickly get to the core!
Frequently asked questions
What do I get when I buy this document?
You get a PDF, available immediately after your purchase. The purchased document is accessible anytime, anywhere and indefinitely through your profile.
Satisfaction guarantee: how does it work?
Our satisfaction guarantee ensures that you always find a study document that suits you well. You fill out a form, and our customer service team takes care of the rest.
Who am I buying these notes from?
Stuvia is a marketplace, so you are not buying this document from us, but from seller luzlinkuz. Stuvia facilitates payment to the seller.
Will I be stuck with a subscription?
No, you only buy these notes for $13.49. You're not tied to anything after your purchase.
Can Stuvia be trusted?
4.6 stars on Google & Trustpilot (+1000 reviews)
67096 documents were sold in the last 30 days
Founded in 2010, the go-to place to buy study notes for 14 years now