What is the difference between metric spaces and vector spaces?

Solution 1:

No, a metric space does not have any particular distinguished point called "the origin". A vector space does: it is defined by the property $0 + x = x$ for every $x$.

In general, in a metric space you don't have the operations of addition and scalar multiplication that you have in a vector space. On the other hand, in general a vector space does not have a notion of "distance".

Solution 2:

Vector spaces necessarily have a vector called the "zero vector"; in the special case of the vector space $k^n$ (where $k$ is a field), this vector is often called "the origin", since $k^n$ also can be seen as a geometric object (the $n$-dimensional affine space). But vector spaces don't necessarily have something we call "the origin": the collection of all polynomials with real coefficients is a real vector space, but we don't normally refer to the zero polynomial as "the origin", even though it is the zero vector of this vector space.

Metric spaces are sets with a metric defined on them. For example, the collection of all complex numbers with complex norm $1$, and with metric given by the usual distance between them as complex numbers, is a metric space. Any nonempty subset of the real numbers, with the usual distance function, is a metric space; and any nonempty set $X$, with distance defined by $d(x,y) = 0$ if $x=y$ and $d(x,y)=1$ if $x\neq y$, is a metric space. There need not be anything that we can reasonably call "the origin."

Solution 3:

A metric space is a set with a distance. That's all you know. That means the set may not have an algebraic structure. for example, {chair, apple} is a set. define d(apple, apple) = d(chair, chair) = 0 and d(chair, apple) = d(apple, chair) = 1. That's a metric space, and it doesnt look like a vectorial space at all.

Solution 4:

I think the OP is confusing a vector space with a normed vector space,which indeed shares many properties of general metric spaces.And for a very simple reason: A norm induces a metric on the underlying set on which the map is defined.This is fairly simple to prove from the definitions and the questioner should try and do it. A general vector space does NOT necessarily have these properties,of course.It is not even necessary that a general vector space admits any notion of distance whatsoever.