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:
“The shortest route is not the most direct one, but rather the one where the most favorable winds swell our sails:Mthat is the lesson that seafarers teach. Not to abide by this lesson is to be obstinate: here, firmness of character is tainted with stupidity.”
—Friedrich Nietzsche (1844–1900)
“What God abandoned, these defended,
And saved the sum of things for pay.”
—A.E. (Alfred Edward)
“Don’t you think I’ve had enough excitement for one evening, without the additional thrill of a strange man making love to me?”
—John L. Balderston (1899–1954)
“... the structure of our public morality crashed to earth. Above its grave a tombstone read, “Be tolerant—even of evil.” Logically the next step would be to say to our commonwealth’s criminals, “I disagree that it’s all right to rob and murder, but naturally I respect your opinion.” Tolerance is only complacence when it makes no distinction between right and wrong.”
—Sarah Patton Boyle, U.S. civil rights activist and author. The Desegregated Heart, part 2, ch. 2 (1962)