# Vector space, norm and metric

This is a follow up question to my previous question.

It is perfectly clear to me what a metric space is. Basically, one can see that notions like convergence and continuity only depend on some sort of closeness or distance, so we can define a distance and then talk about these things.

Now it might happen that we have a vector space and we would like to talk about these concepts. Now in some cases we want the metric to respect the vector space operations because otherwise we can't really make use of them.

This means that the distance should have some additional properties: $$(i) d(w,v)=d(w+u,v+u)$$ $$(ii) d(\alpha u,\alpha v)=\lvert\alpha\rvert d(u,v)$$

In most cases we introduce a norm which in turn induces a metric with these properties. The norm itself can be seen as a generalization of length, size or magnitude, so it is also an inuitive concept.

Now I've got 2 questions:

1)I am wondering if the only reason to introduce a norm is to be able to talk about convergence and continuity or are there other reasons why one would want a generalization of length for arbitrary vector spaces. Are there other things we can do once we have a norm? If yes, please provide an example. If not why don't we just define a norm as a metric with the additional properties (i) and (ii). I am guessing it´s because the properties of the norm are sufficient and we want our definitions to be short.

2) What is the intuition for the triangle inequality for an arbitrary norm in any vector space? In $$\mathbb{R^{n}}$$ it makes sense to require that the length of a sum of two vectors is smaller than the sum of the length since the directions might be different. For example, why should this be true if I have some norm for a function space? For a distance this is intuitive since it basically says that the shortest path between 2 points is a line between the points.

Thanks very much!

Edit:

Regarding 2) it might make sense to view the norm as a generalization of the absolute value rather than the length in $$\mathbb{R^{n}}$$. Intuitively the absolute value gives the magnitude of a number. The triangle inequality can be seen as the property of subadditivity. In higher dimensions it then makes sense to say the magnitude of a vector is its length. Of course it satisfies the properties of a norm.

• I think you are using the word "generalisation" back to front
– Ali
Feb 12, 2020 at 11:22
• I am not sure what you mean. But I've just corrected an error. I meant to say "The norm itself can be seen as a generalization of length, size or magnitude...". Feb 12, 2020 at 11:27
• I've only said that a norm induces a metric. The norm itself is not a metric and as I've pointed out it generalizes the notion of length, size or magnitude while a metric is a generalization of distance. I see what you mean, but I am interested in the norm viewed as generalization of length. Feb 12, 2020 at 11:32
• Sorry I think you had said that the norm was a generalisation of the distance
– Ali
Feb 12, 2020 at 11:33
• I guess, condition (ii) would go like $d(\alpha u,\, \alpha v) =\alpha\cdot d(u,v)$ for $\alpha>0$, right? Is it clear that a metric on an $\Bbb R$-vector space is induced by a norm iff (i) and (ii) hold? Feb 12, 2020 at 11:40