Inverse of Metric in Topology: Schwarz Inequality

In summary: However, D(x,z) = 1/2 + 1/3 = 2, so D(x,y) <= D(x,z) + D(y,z), which is satisfied. This shows that D is a metric on the space consisting of the points (0,0), (1,1), and (2,2).
  • #1
Bachelier
376
0
In Topology:

is the multiplicative inverse of a metric, a metric?

How do we define the Schwarz inequality then?

if ##d(x,z) ≤ d(x,y) + d(y,z)## the inverse ##1/d(x,z)## would give the opposite?
 
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  • #2
Hey Bachelier.

Try taking the reciprocal (also flip the inequality) and collect terms.
 
  • #3
chiro said:
Hey Bachelier.

Try taking the reciprocal (also flip the inequality) and collect terms.

Hi Chiro:

I don't know if what you mean is taking the reciprocal of ##d(x,y)##

indeed if I start with ##d(x,z) ≤ d(x,y) + d(y,z)##

then ##\frac{1}{d(x,z)} ≥ \frac{1}{d(x,y) + d(y,z)}##

would lead to ##d(x,z) ≤ d(x,y) + d(y,z)## again.

But what I feel like is the need to show that: my new metric defined as ##D(x,y) = \frac{1}{d(x,y)}## respects the triangle inequality.

##i.e. D(x,z) ≤ D(x,y) + D(y,z)## (result A)

but the way the metric ##D## is defined would give me

##D(x,z) ≥ D(x,y) + D(y,z)## not result A

Failing this will not make ##D## a metric.
 
  • #4
I don't think your D metric will be a metric at all.
 
  • #5
chiro said:
I don't think your D metric will be a metric at all.

I thought so. I heard someone talking about the inverse of a metric as being a metric on the same space.
 
  • #6
Consider the standard euclidean metric on the real numbers d(x,y) = |x-y|, and let D(x,y) = 1/d(x,y). Even overlooking the problem that this is not defined when x=y, we can find a simple counterexample. Let x = 0, y = 3, z = 1. Then D(x,y) = 1/3, D(y,z) = 1/2, and D(x,z) = 1, so the triangle inequality D(x,z) <= D(x,y) + D(y,z) is not satisfied.
 

Related to Inverse of Metric in Topology: Schwarz Inequality

1. What is the Inverse of Metric in Topology?

The inverse of metric in topology refers to a mathematical property that is used to measure the distance between two points in a given space. It is used to define the concept of distance in topology and is closely related to the concept of a metric space.

2. What is the Schwarz Inequality?

The Schwarz Inequality, also known as the Cauchy-Schwarz Inequality, is a fundamental inequality in mathematics that relates the inner product of two vectors in an inner product space to their lengths. In simpler terms, it states that the dot product of two vectors is always less than or equal to the product of their magnitudes.

3. How are the Inverse of Metric and Schwarz Inequality related?

The Inverse of Metric and Schwarz Inequality are related in the sense that the Inverse of Metric is a key component in the proof of the Schwarz Inequality. The Inverse of Metric is used to define the inner product in an inner product space, which is essential in proving the inequality.

4. Why is the Inverse of Metric important in Topology?

The Inverse of Metric is important in Topology because it allows us to define the concept of distance in a topological space. This is crucial in studying the properties of topological spaces and understanding their structures. It also plays a key role in proving important theorems and inequalities, such as the Schwarz Inequality.

5. How is the Inverse of Metric used in practical applications?

The Inverse of Metric has practical applications in many fields, including physics, engineering, and computer science. It is used to define distance and similarity measures in data analysis and pattern recognition. It is also used in optimization problems and in the study of geometric structures in physics and engineering.

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