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:
“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)
“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)