Properties
The operator norm is indeed a norm on the space of all bounded operators between V and W. This means
The following inequality is an immediate consequence of the definition:
The operator norm is also compatible with the composition, or multiplication, of operators: if V, W and X are three normed spaces over the same base field, and A : V → W and B: W → X are two bounded operators, then
For bounded operators on V, this implies that operator multiplication is jointly continuous.
It follows from the definition that a sequence of operators converge in operator norm means they converge uniformly on bounded sets.
Read more about this topic: Operator Norm
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)