Shouldn't this definition of a metric include a square root?

In summary: The conversation discusses the relationship between inner product spaces and metric spaces. The definition of a metric is given in both Mathworld and Wikipedia as it relates to inner product spaces. However, there appears to be a difference in the definition of the metric given by both sources. In summary, the Mathworld definition is the square of the Wikipedia definition, leading to a violation of the triangle inequality. This suggests a mistake on Wolfram's part.
  • #1
nomadreid
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TL;DR Summary
Mathworld.Wolfram says that there is a metric on an inner product space (with inner product <.,.>) defined by <v-w,v-w>. Shouldn't that be the square root of <v-w,v-w>?
In https://mathworld.wolfram.com/InnerProduct.html, it states
"Every inner product space is a metric space. The metric is given by
g(v,w)= <v-w,v-w>."
In https://en.wikipedia.org/wiki/Inner_product_space , on the other hand,
"As for every normed vector space, an inner product space is a metric space, for the distance defined by
d(x,y) = ||y-x||"
after it had defined ||x|| as sqrt(<x,x>)

(I am assuming that a metric is the same as a distance function.)

The Mathworld definition appears to be the square of the Wikipedia definition. Whereas one can have more than one metric on a vector space, the former definition would seem to violate the triangle inequality
[Example: take the inner product on the one-dimensional vector space (the number line) as the dot product (which reduces to multiplication of the coordinates of the representative vectors from the origin), and check the triangle inequality for the a=0, b=2, c=5: 4+9 < 25. ]

Did Wolfram make a mistake, or am I missing some elementary and obvious point?
 
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  • #2
nomadreid said:
Did Wolfram make a mistake?
It looks like it.
 
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  • #3
I haven't checked the triangle identity, but isn't the square of the Euclidean metric a metric, too?
 
  • #4
fresh_42 said:
I haven't checked the triangle identity, but isn't the square of the Euclidean metric a metric, too?
It doesn't obey the triangle inequality.
 
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  • #5
It looks metric is a general term, while distance is specific.
 
  • #7
The only difference is, that metric is a technical term, and distance is a kind of interpretation. E.g. if we consider an ##L^2## space of functions, then we have a metric, and therewith a distance. However, there is no intuition of the distance between two functions, so people speak about norms and metrics.
 
  • #8
Sometimes the term "distance" is also used as a technical term outside of the intuitive Euclidean distance: for example, the Hamming distance (which is a metric).
 
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  • #9
nomadreid said:
Sometimes the term "distance" is also used as a technical term outside of the intuitive Euclidean distance: for example, the Hamming distance (which is a metric).
I constructed a problem in one of my challenge threads that was the theorem of Thales disguised in the language of ##L^2## spaces.
 

Related to Shouldn't this definition of a metric include a square root?

1. What is a metric in science?

A metric in science is a quantitative measure used to describe and compare different aspects of a phenomenon or system. It is typically expressed as a numerical value and is used to evaluate and understand the relationships between different variables.

2. Why is the square root often included in the definition of a metric?

The square root is often included in the definition of a metric because it allows for the measurement of a quantity's magnitude, rather than just its direction. This is particularly useful when dealing with physical quantities that have both positive and negative values, such as velocity or temperature.

3. Shouldn't this definition of a metric include a square root?

Whether or not a square root should be included in the definition of a metric depends on the specific context and purpose of the measurement. In some cases, a square root may be necessary to accurately represent the quantity being measured, while in others it may not be relevant or appropriate.

4. Can a metric be defined without including a square root?

Yes, a metric can be defined without including a square root. In fact, there are many metrics that do not involve square roots at all, such as the mean, standard deviation, or correlation coefficient. The inclusion of a square root is not a requirement for a measurement to be considered a metric.

5. How does the inclusion of a square root affect the interpretation of a metric?

The inclusion of a square root can affect the interpretation of a metric in several ways. It can change the scale and range of the metric, making it more or less sensitive to changes in the underlying data. It can also impact the mathematical relationships between different metrics, making it important to consider the context and purpose of the measurement when interpreting its results.

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