Metric spaces RXR how to prove?

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In summary, the discussion focused on the Euclidean metric on the real line (R) and the set of complex numbers (C). The participants were asked to show whether the Euclidean metric is a metric for R x R and C x C, and to generalize the metric to a set of n-tuples of real numbers, R^n. The question was also raised about whether this generalization holds for the set of n-tuples of complex numbers, C^n.
  • #1
fabbi007
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R is real line, C is set of Complex numbers

If we considered the Euclidean metric on RXR

a. Show whether the Euclidean metric on R
RXR is a metric.
b. Show whether the Euclidean metric on C
C is a metric.
c. Generalize the Euclidean metric to a set made up of all n-tuples of real numbers
X=R power n. Is this also true on C power n?

Can I get some direction on solving this proofs?
 
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  • #2
a. Show whether the Euclidean metric on RxR is a metric.

There is a definition for "metric". Check all 3 clauses of that definition in this particular case.
 

1. What is a metric space?

A metric space is a mathematical concept that describes a set of objects where the distance between any two objects is well-defined. It consists of a set of objects, called points, and a function that assigns a non-negative real number to pairs of points, known as the distance function.

2. How do you prove that a set is a metric space?

To prove that a set is a metric space, you need to show that it satisfies the three axioms of a metric space: non-negativity, symmetry, and the triangle inequality. This means that the distance between any two points must be a non-negative real number, the distance between a point and itself is zero, and the distance between any three points satisfies the triangle inequality.

3. What is the significance of the RXR notation in metric spaces?

The RXR notation in metric spaces refers to the Cartesian product of two sets, where R represents the set of real numbers. This notation is used to indicate that the points in the metric space are ordered pairs of real numbers, and the distance between them is calculated using the standard Euclidean distance formula.

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

One example of a metric space is the set of all points on a line, where the distance between any two points is the absolute value of their difference. Another example is the set of all points in a plane, where the distance between two points is calculated using the Pythagorean theorem.

5. How can you prove that two metric spaces are isometric?

To prove that two metric spaces are isometric, you need to show that there exists a bijective function between the two spaces that preserves the distance between any two points. This means that the distance between two points in one space is equal to the distance between their corresponding points in the other space.

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