Direct Sum of Modules With Additional Structure
If the modules we are considering carry some additional structure (e.g. a norm or an inner product), then the direct sum of the modules can often be made to carry this additional structure, as well. In this case, we obtain the coproduct in the appropriate category of all objects carrying the additional structure. Three prominent examples occur for algebras over a field, Banach spaces and Hilbert spaces.
Read more about this topic: Direct Sum Of Modules
Famous quotes containing the words direct, sum, additional and/or structure:
“I should say that the most prominent scientific men of our country, and perhaps of this age, are either serving the arts and not pure science, or are performing faithful but quite subordinate labors in particular departments. They make no steady and systematic approaches to the central fact.... There is wanting constant and accurate observation with enough of theory to direct and discipline it. But, above all, there is wanting genius.”
—Henry David Thoreau (1817–1862)
“To sum up:
1. The cosmos is a gigantic fly-wheel making 10,000 revolutions a minute.
2. Man is a sick fly taking a dizzy ride on it.
3. Religion is the theory that the wheel was designed and set spinning to give him the ride.”
—H.L. (Henry Lewis)
“Dog. A kind of additional or subsidiary Deity designed to catch the overflow and surplus of the world’s worship.”
—Ambrose Bierce (1842–1914)
“There is no such thing as a language, not if a language is anything like what many philosophers and linguists have supposed. There is therefore no such thing to be learned, mastered, or born with. We must give up the idea of a clearly defined shared structure which language-users acquire and then apply to cases.”
—Donald Davidson (b. 1917)