Finite Product Spaces
Given n seminormed spaces Xi with seminorms qi we can define the product space as
with vector addition defined as
and scalar multiplication defined as
- .
We define a new function q
for example as
- .
which is a seminorm on X. The function q is a norm if and only if all qi are norms.
More generally, for each real p≥1 we have the seminorm:
For each p this defines the same topological space.
A straightforward argument involving elementary linear algebra shows that the only finite-dimensional seminormed spaces are those arising as the product space of a normed space and a space with trivial seminorm. Consequently, many of the more interesting examples and applications of seminormed spaces occur for infinite-dimensional vector spaces.
Read more about this topic: Normed Vector Space
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