Bounded Operator
In functional analysis, a branch of mathematics, a bounded linear operator is a linear transformation L between normed vector spaces X and Y for which the ratio of the norm of L(v) to that of v is bounded by the same number, over all non-zero vectors v in X. In other words, there exists some M > 0 such that for all v in X
The smallest such M is called the operator norm of L.
A bounded linear operator is generally not a bounded function; the latter would require that the norm of L(v) be bounded for all v, which is not possible unless Y is the zero vector space. Rather, a bounded linear operator is a locally bounded function.
A linear operator on a metrizable vector space is bounded if and only if it is continuous.
Read more about Bounded Operator: Examples, Equivalence of Boundedness and Continuity, Linearity and Boundedness, Further Properties, Properties of The Space of Bounded Linear Operators, Topological Vector Spaces
Famous quotes containing the word bounded:
“I could be bounded in a nutshell and count myself a king of
infinite space, were it not that I have bad dreams.”
—William Shakespeare (1564–1616)