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:
“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)