Infinite Dimensions
Note that in an infinite dimensional space, we can have a bilinear form ƒ for which is injective but not surjective. For example, on the space of continuous functions on a closed bounded interval, the form
is not surjective: for instance, the Dirac delta functional is in the dual space but not of the required form. On the other hand, this bilinear form satisfies
- for all implies that
Read more about this topic: Degenerate Form
Famous quotes containing the words infinite and/or dimensions:
“The universe is then one, infinite, immobile.... It is not capable of comprehension and therefore is endless and limitless, and to that extent infinite and indeterminable, and consequently immobile.”
—Giordano Bruno (15481600)
“Words are finite organs of the infinite mind. They cannot cover the dimensions of what is in truth. They break, chop, and impoverish it.”
—Ralph Waldo Emerson (18031882)