What is the motivation behind the definition of bounded set in a topological vector space? The definition is different from the boundedness definition in metric space.
Why is it not simply defined as follows: A set is bounded if there is a neighbourhood around zero which contains that set ?
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A bounded set $S$ in a topological vector space is one for which, given any neighborhood $N$ of the zero vector, there is a scalar $\alpha$ such that $S \subset \alpha N$. For more: http://en.wikipedia.org/wiki/Bounded_set_(topological_vector_space)
The motivation for this definition is fairly straightforward--if a "stretching" of a neighborhood of the zero vector contains the set, it is, in that sense, "bounded". In this way, the definition of boundedness is made not to depend upon distance, but solely on the notions of scaling, and of open sets.
In a metric space you have a way to measure distance and thus the definition of boundedness is, in a sense, obvious. In a topological vector space there are no distances, so of course the definition will be different than that given for metric spaces simply since you can't state the metric definition without a metric. However, if you think about it, the intuition is still the same. Try contemplating the definition. Stare at it for a while, translate it back to a familiar (metric!) topological vector space until you realize that it says essentially the same thing only using a more general language, since that is all you have.