Properties
- Any linear functional L is either trivial (equal to 0 everywhere) or surjective onto the scalar field. Indeed, this follows since just as the image of a vector subspace under a linear transformation is a subspace, so is the image of V under L. But the only subspaces (i.e., k-subspaces) of k are {0} and k itself.
- A linear functional is continuous if and only if its kernel is closed (Rudin 1991, Theorem 1.18).
- Linear functionals with the same kernel are proportional.
- The absolute value of any linear functional is a seminorm on its vector space.
Read more about this topic: Linear Functional
Famous quotes containing the word properties:
“The reason why men enter into society, is the preservation of their property; and the end why they choose and authorize a legislative, is, that there may be laws made, and rules set, as guards and fences to the properties of all the members of the society: to limit the power, and moderate the dominion, of every part and member of the society.”
—John Locke (1632–1704)
“A drop of water has the properties of the sea, but cannot exhibit a storm. There is beauty of a concert, as well as of a flute; strength of a host, as well as of a hero.”
—Ralph Waldo Emerson (1803–1882)