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 process of writing has something infinite about it. Even though it is interrupted each night, it is one single notation.”
—Elias Canetti (b. 1905)
“I was surprised by Joe’s asking me how far it was to the Moosehorn. He was pretty well acquainted with this stream, but he had noticed that I was curious about distances, and had several maps. He and Indians generally, with whom I have talked, are not able to describe dimensions or distances in our measures with any accuracy. He could tell, perhaps, at what time we should arrive, but not how far it was.”
—Henry David Thoreau (1817–1862)