Vector Spaces, Normed Vector Spaces and Metric spaces I've already studied real analysis and I've just finished studying linear algebra (the source I've used did not cover norms, but I have some basic understanding about them).
Now I know that there are normed vector spaces and they have a lot of applications. From my understanding the reason for defining them is that it is a way to give a vector space some additional structure to be able to consider things like convergence and continuity. This is because a norm induces a metric, and therefore all the metric space theorems are applicable.
Now I've got two questions:
1) Although I can mathematically understand that a norm induces a metric and it also intuitively makes sense in euclidean spaces since the norm can be interpreted as length which makes the connection to the metric or distance obvious (We can just draw two vectors in $\mathbb{R}^{2}$ and then it is easy to see the that the relation follows by the Pythagorean Theorem.) However, I was wondering why this holds for any normed vector space. In general, the norm can be seen as magnitude or size of an object while the metric measures similarity. Can someone give me an intuition about the connection between norm and metric in a broader context?
2) As mentioned above the ultimate goal of defining the norm is to introduce a metric space structure. I've read different posts on this topic and it seems that we want "the metric space structure to play nice with the vector space structure" (Metric spaces and normed vector spaces). Can someone give me an example of an application where this goes wrong and what the consequences are? Translation invariance and homogeneity seem to be important properties for this (What's the need of defining notion of distance using norm function in a metric space?).
 A: 
However, I was wondering why this holds for any normed vector space. In general, the norm can be seen as magnitude or size of an object while the metric measures similarity. Can someone give me an intuition about the connection between norm and metric in a broader context?

If you can measure the size of an object and you can subtract objects, then you can produce a measure of similarity. More precisely, if $\|\cdot\|$ is a norm (measure of size), then your measure of similarity is the "size of the difference", i.e.
$$
d(x,y) = \|x-y\|.
$$

We want "the metric space structure to play nice with the vector space structure". Can someone give me an example of an application where this goes wrong and what the consequences are?

Here is an example of a metric on $\Bbb R$.  We define
$$
d(x,y) = \begin{cases}
0 & x=y\\
\min\{|x-y|,1\} & x=0 \text{ or } y = 0\\
1 & \text{otherwise}
\end{cases}
$$
This defines a metric. The difficult thing to prove here is the triangle inequality when $x=0$ but $y,z$ are non-zero; we find
$$
\min\{|z|,1\} = d(x,z) \leq d(x,y) + d(y,z) = \min\{|y|,1\} + 1.
$$
Here's something that goes wrong: we would expect that for $f:(\Bbb R, d) \to (\Bbb R,|\cdot|)$ and any $c \in \Bbb R$, $f(x - c)$ is continuous if and only if $f(x)$ is continuous.  However, this is not the case.
A: 1) Starting from a normed vector space $V$, then if $v\in V$ we write the norm as $\|v\|$ and it should be thought of as the magnitude of the vector $v$, i.e. the distance from the origin. Now the vector space has some symmetries which we want the metric to preserve. Think of translation in Euclidean space: if we shift the two objects we are comparing in the same way, their distance stays the same. In an arbitrary vector space, the condition on the metric is that $d(a+x,b+x)=d(a,b)$. If we choose $x=-b$ then we get $d(a,b)=d(a-b,0)$. But we already said that the norm $\|x\|$ is the distance from the origin, i.e. $d(x,0)$, so this means the metric must be given by $d(a,b)=d(a-b,0)=\|a-b\|$.
2) In general nothing "goes wrong", it depends on the application, although one of the lessons of physics is that losing symmetries is not something that should be done lightly.
