Normed Vector Space - Finite Product Spaces

Finite Product Spaces

Given n seminormed spaces X_i with seminorms q_i 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 q_i 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.