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
Famous quotes containing the words finite, product and/or spaces:
“For it is only the finite that has wrought and suffered; the infinite lies stretched in smiling repose.”
—Ralph Waldo Emerson (1803–1882)
“Evil is committed without effort, naturally, fatally; goodness is always the product of some art.”
—Charles Baudelaire (1821–1867)
“Though there were numerous vessels at this great distance in the horizon on every side, yet the vast spaces between them, like the spaces between the stars,—far as they were distant from us, so were they from one another,—nay, some were twice as far from each other as from us,—impressed us with a sense of the immensity of the ocean, the “unfruitful ocean,” as it has been called, and we could see what proportion man and his works bear to the globe.”
—Henry David Thoreau (1817–1862)