Euclidean norm gives length even in $>3$ dimensions? In $1,2,3$ dimensions I can simply make triangles and see that Euclidean norm gives me the distance between two points (i.e. the length of the vector from one point to the other). In higher dimensions, though, how do we know that the euclidean norm still represents a length? It makes sense that we could still decompose the vector between the points into multiple triangles, but how are we sure that things don't become weird -- i.e. decomposition into triangles becomes invalid -- in $4,5,6$ dimensions?
In other words, I am wondering if there is either a proof (or if someone can supply a proof), or just some really good intuition why the euclidean norm continues to be a way of finding the length between two points in higher dimensions.
 A: As comments already pointed out, you can't have a physical space with more than 3 spatial dimensions. So the concept of length as something which can be measured with a ruler breaks down. You can simply continue to use the term “length” for the norm, since it helps intuition, it is compatible with everyday experience in lower dimensions and there is no higher-dimension reality which could conflict with this. But it's essentially just a colloquial alias for “norm”.
Regarding your triangles considerations: for any dimension greater than one, one can compute the length/norm by breaking the distance down into one component parallel to one of the coordinate directions, and one component perpendicular to it. The former is a one-dimensional length, the latter a $(n-1)$-dimensional one, so you can recurse until you end up in a space of dimension $\le3$. The problem with this is that the concept of orthogonality in higher dimensions has about as much justification as that of length: we simply call things orthogonal if the dot product is zero, even though we couldn't verify it's actually perpendicular by holding a set square to it.
All of the above assumes a simple $\mathbb R^n$. There are spaces which have very different ideas of length, distance and/or norm. The comment by user254665 points out some such spaces, which I repeat with links to Wikipedia and some summary:


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*A metric space has a defined way to measure distances, not neccessarily using a norm. Perhaps it would be more correct above to say that “length” is a colloquial term for “distance as computed by the metric”, but for everiday applications it's the same.

*A normed vectorspace has a defined norm, which in turn implies a metric.

*An inner product space is a vector space which comes with a defined inner product. There the Cauchy–Schwarz inequality holds, which is a generalized triangle inequality.

*A Hilbert space is an inner product space which is complete with respect to the metric induced by that inner product.

