Some a questions in Metric Spaces

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In summary: These questions are asking for proofs of various properties in metric spaces. In summary, the conversation is discussing how to prove certain properties in metric spaces, such as boundedness of Cauchy sequences, using specific definitions and postulates. The speaker also mentions needing help with speed.
  • #1
cotton candy
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Hii
my Dears...

I'm new student in the Math & I'm so bad in the English Language..
But, I want to learn this language ...


to excuse me ...

I have some a questions about Metric Spaces ..

Q1:If (X,d) is a metric spaces . Prove the fallwing:

1* ld(x,y)-d(z,y)l [tex]\leq[/tex] d(x,y)+d(y,w).
2* ld(x,z)-d(y,z)l [tex]\leq[/tex] d(x,y). ..??



Q2:Prove that:
Xn ـــــــــ> X iff [tex]\forall[/tex] V (neighborhood of X) [tex]\exists[/tex] n0 is number s.t Xn[tex]\in[/tex]V [tex]\forall[/tex]n>n0...??


Q3: If (X1,d1) & (X2,d2) is a metrics spaces, Prove that X=X1xX2 () is a metric spaces whith a metric defind by: d(x,y)=d1(x1,y1)+d2(x2,y2) s.t x1,y1[tex]\in[/tex]X1..??


Q4:Prove that every Cauchy sequence in a metric space (X, d) is bounded...??


I need to help by speed..
Thanx ...
 
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  • #2
cotton candy said:
Hii
my Dears...

I'm new student in the Math & I'm so bad in the English Language..
But, I want to learn this language ...


to excuse me ...

I have some a questions about Metric Spaces ..

Q1:If (X,d) is a metric spaces . Prove the fallwing:

1* ld(x,y)-d(z,y)l [tex]\leq[/tex] d(x,y)+d(y,w).
2* ld(x,z)-d(y,z)l [tex]\leq[/tex] d(x,y). ..??
Using what basic definitions, postulates, etc.?



Q2:Prove that:
Xn ـــــــــ> X iff [tex]\forall[/tex] V (neighborhood of X) [tex]\exists[/tex] n0 is number s.t Xn[tex]\in[/tex]V [tex]\forall[/tex]n>n0...??
What is your definition of "Xn ـــــــــ>X"?


Q3: If (X1,d1) & (X2,d2) is a metrics spaces, Prove that X=X1xX2 () is a metric spaces whith a metric defind by: d(x,y)=d1(x1,y1)+d2(x2,y2) s.t x1,y1[tex]\in[/tex]X1..??
Show that the conditions for a metric space are satisfied- in other words what is the definition of "metric space".


Q4:Prove that every Cauchy sequence in a metric space (X, d) is bounded...??
If {xn} is a Cauchy sequence, then there exist N such that if both m, n> N, d(xn, xm)< 1. Let M= largest of d(xn,xm) for n and m [itex]\le[/itex] N+1. Can you prove that d(xn,xm)[itex]\le[/itex] M+ 1 for all m and n.

I need to help by speed..
Thanx ...[/QUOTE]

In mathematics, definitions are working definitions. You use the specific words of definitions in proofs.
 

Related to Some a questions in Metric Spaces

1. What is a metric space?

A metric space is a mathematical structure that consists of a set of elements and a distance function that measures the distance between any two elements in the set. This distance function satisfies certain properties, such as being non-negative, symmetric, and satisfying the triangle inequality.

2. How is a metric space different from a Euclidean space?

A Euclidean space is a type of metric space where the distance function is based on the Pythagorean theorem. However, a metric space can have a different distance function that does not necessarily follow the rules of Euclidean geometry.

3. What is the importance of metric spaces in mathematics?

Metric spaces are important in mathematics because they provide a way to define and study the concept of distance between objects, which is a fundamental concept in many areas of mathematics such as topology, analysis, and geometry. They also have many practical applications in areas such as computer science, physics, and engineering.

4. Can you give an example of a metric space?

One example of a metric space is the set of real numbers with the distance function defined as the absolute difference between two numbers. Another example is the set of all continuous functions on a closed interval with the distance function defined as the maximum absolute difference between two functions.

5. What are some common properties of metric spaces?

Some common properties of metric spaces include the non-negativity of the distance function, the symmetry of the distance function, and the triangle inequality, which states that the distance between any two points in the metric space is always less than or equal to the sum of the distances between those points and a third point.

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