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:
“Sisters define their rivalry in terms of competition for the gold cup of parental love. It is never perceived as a cup which runneth over, rather a finite vessel from which the more one sister drinks, the less is left for the others.”
—Elizabeth Fishel (20th century)
“The UN is not just a product of do-gooders. It is harshly real. The day will come when men will see the UN and what it means clearly. Everything will be all right—you know when? When people, just people, stop thinking of the United Nations as a weird Picasso abstraction, and see it as a drawing they made themselves.”
—Dag Hammarskjöld (1905–1961)
“Every true man is a cause, a country, and an age; requires infinite spaces and numbers and time fully to accomplish his design;—and posterity seem to follow his steps as a train of clients.”
—Ralph Waldo Emerson (1803–1882)