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Well, I was not quite aware that addition of points is not defined in metric spaces but is defined only on linear spaces and others.

Could anyone elaborate why is this? Is the addition of intervals (closed or open) defined?

I am confused as to how addition of points is undefined on metric space but is defined on linear space.

I would appreciate if anyone clears this misconception/confusion.

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4 Answers 4

up vote 5 down vote accepted

Simply put: a given metric space needn't have the structure necessary to perform addition.

A vector space (linear space) $V$ comes equipped with all sorts of structure: addition, scalar multiplication, distributivity laws, a distinguished 'zero' (i.e. the origin), etc. These are defined by functions which map back into the space. For instance, addition can be seen as a map $V \times V \to V$, taking a pair $(v,w)$ to another element of $V$ which we denote by $v+w$.

However, in a general metric space $X$, this doesn't happen. The only structure you have is the metric, which is a function $d: X \times X \to [0,\infty)$. The metric allows you to relate a given pair of elements of the space, by measuring the 'distance' between them. But it doesn't in general allow you to distinguish a special point, and it doesn't allow you to perform operations on elements of the space that spit out elements of the space (rather than numbers).

Added: The notion of an interval can be generalised to metric spaces by introducing balls. The (open) ball of radius $r > 0$ about a point $x \in X$ is a set $$B(x;r) = \{ y \in X\, :\, d(x,y) < r \} \subseteq X$$ i.e. the set of points in $X$ which lie at a distance $<r$ from $x$. Likewise, the closed ball of radius $r$ about $x$ is given by $$\bar B(x;r) = \{ y \in X\, :\, d(x,y) \le r \} \subseteq X$$ You can check that these really do give you intervals when $X=\mathbb{R}$ with the usual metric. I presume this is what you were asking when you mentioned "addition of intervals".

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In the definition of a vector space, there is the operation of vector addition where two vectors are added. In the definition of a metric space, there is no addition. Whether the addition of intervals is defined depends on whether we have defined such a thing.

The same set can become different metric spaces if we define different metrics on it. To talk about addition in a metric space, one needs to define extra structure on it which has nothing to do with the metric itself.

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Why was this downvoted? (+1) –  Clive Newstead Jan 3 '13 at 16:13

A metric space is a set $X$ together with a metric $d$. The elements of $X$ may also be called points in $X$. The points do not have to be real numbers so there is not necessarily a predefined notion of what it means to add them. There is also no predefined notion of what an interval is. You see, the notion of an interval of one of order. Which elements are between $a$ and $b$? The answer is an interval. On a metric space there is not necessarily a notion of what order the elements come in, so the notion of an interval does not make sense.

To contrast, when defining a linear space (vector space), in the prescription of the definition of the space one must define the operator $+$ on elements in the space. That is, in order to call something a vector space, you must be able to describe to me a method adding them which is consistent with the axioms of addition. Though even in a vector space there is not necessarily a notion of order.

Misconception may arise if the only example of a metric space that you have seen is the real line. The wiki http://en.wikipedia.org/wiki/Metric_space has plenty of examples of metric spaces which are not the real line. You may want to look into these.

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In metric spaces, we have a notion of distance because by definition, a metric space is a space with a metric. But a priori there is no notion of addition. Simply because it does not appear as part of the concept.

In abelian groups, we have a notion of addition, again by the very definition of the concept of abelian group. But a priori no notion of distance.

A vector space is an abelian group with additional structure (you can multiply with elements of the ground field in a consistant manner). So we still have addition, but no metric.

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