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

• I
Gold Member
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?

PeroK
Homework Helper
Gold Member
2020 Award
Did Wolfram make a mistake?
It looks like it.

fresh_42
Mentor
I haven't checked the triangle identity, but isn't the square of the Euclidean metric a metric, too?

PeroK
Homework Helper
Gold Member
2020 Award
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.

mathman
It looks metric is a general term, while distance is specific.

fresh_42
Mentor
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.

Gold Member
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).

fresh_42
fresh_42
Mentor
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.