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I'm now studying metric space. Here, I don't understand why definitions of distance and norm in euclidean space are repectively given in my book.

I understand the difference between two concepts when i'm working on non-euclidean space, but is there any even slight difference between these two concepts when it is $\mathbb{R}^k$?

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  • $\begingroup$ What definitions are you referring to? $\endgroup$
    – M Turgeon
    Commented Jul 17, 2012 at 17:05

6 Answers 6

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All norms can be used to create a distance function as in $d(x,y) = \|x-y\|$, but not all distance functions have a corresponding norm, even in $\mathbb{R}^k$. For example, a trivial distance that has no equivalent norm is $d(x,x) = 0$, and $d(x,y) = 1$, when $x\neq y$. Another distance on $\mathbb{R}$ that has no equivalent norm is $d(x,y) = | \arctan x - \arctan y|$.

However, in general, when working in $\mathbb{R}^k$ the distance used is one induced by a norm, and 'unusual' distances are typically used to illustrate other mathematical concepts (eg, the $\arctan$ distance gives an example of an incomplete metric space).

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What user29999 said was the main difference, i.e.: a distance is a function

$$d:X \times X \longrightarrow \mathbb{R}_+$$

while a norm is a function:

$$\| \cdot \| X \longrightarrow \mathbb{R}_+$$

However, I think that you wonder whether one induces the other. So a norm always induces a distance by:

$$d(x,y) = \|x-y\|$$

However, the other way around is not always true. For a distance to come from a norm, it needs to verify:

$$d(\alpha x, \alpha y) = |\alpha | d(x,y)$$

If we take the discrete distance on any space:

$$d(x,y) = \begin{cases} 0, \text{ if $x = y$}\\ 1, \text{ if $x \ne y$} \end{cases}$$

Then this distance does not verify the condition, e.g. for $\alpha = 2$.

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You can take the norm of one element. A distance needs two elements. Hence we cannot talk about the distance of an element.

For example: The absolute value on the real numbers is a norm. For example $\lvert -3 \lvert = 3$. The corresponding distance is $d(x,y) = \lvert x - y\lvert$. For example $d(-3, 7) = \lvert -3 - 7\lvert = \lvert -10\lvert = 10$.

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    $\begingroup$ ...but we can trivially define the distance for a single element in a vector space as the distance to the origin. $\endgroup$ Commented Apr 23, 2015 at 0:34
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The distance is a two vectors function $d(x,y)$ while the norm is a one vector function $||v||$. However, frequently you use the norm to calculate the distance by means of the difference of two vectors $||y-x||$.

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  • $\begingroup$ "A norm is a generalization of the intuitive notion of "length" in the physical world." in Normed vector space $\endgroup$
    – JeeyCi
    Commented Mar 31 at 12:41
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Distance is a function $d:X\times X \longrightarrow \mathbb{K}$ and $Norm$ is a function $n:X \longrightarrow \mathbb{K}$ where $\mathbb{K}$ is a field.

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The metric space/distance metric is a mathematical construction built up on topological space (definition of convergence, continuity, completeness, separability, connectedness, compactness etc. etc.). The distance operates with point sets and doesn't know the vectors. The Hilbert space is built up on metric space and defines new operation as inner product, vectors and using norms. The metric space itself doesn't know norms and vectors, since as is introduced on Hilbert space. Below is a derivation how to utilize metric space in Hilbert space. enter image description here

Reference: Muscat, J. Functional Analysis An Introduction to Metric Spaces, Hilbert Spaces, and Banach Algebras, Springer, 2014.

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