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:
“...if you are to gain any great amount of good from the world, you must attain a passive condition of mind. ...it is never to be forgotten that it is the rest of the world and not you that holds the great share of the world’s wealth, and that you must allow yourself to be acted upon by the world, if you would become a sharer in the gain of all the ages to your own infinite advantage.”
—Anna C. Brackett (1836–1911)
“The truth is that a Pigmy and a Patagonian, a Mouse and a Mammoth, derive their dimensions from the same nutritive juices.... [A]ll the manna of heaven would never raise the Mouse to the bulk of the Mammoth.”
—Thomas Jefferson (1743–1826)