Difference between Norm and Distance

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 $\mathbb{R}^k$?

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).

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 $\alpha = 2$.

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 $d(-3, 7) = \lvert -3 - 7\lvert = \lvert -10\lvert = 10$.

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.