In functional analysis and operator theory, a bounded linear operator is a linear transformation
between topological vector spaces (TVSs)
are normed vector spaces (a special type of TVS), then
A linear operator between normed spaces is continuous if an only if it is bounded.
The concept of a bounded linear operator has been extended from normed spaces to all topological vector spaces.
is called "bounded" then this usually means that its image
A linear map has this property if and only if it is identically
Consequently, in functional analysis, when a linear operator is called "bounded" then it is never meant in this abstract sense (of having a bounded image).
Every bounded operator is Lipschitz continuous at
A linear operator between normed spaces is bounded if and only if it is continuous.
Conversely, it follows from the continuity at the zero vector that there exists a
between two topological vector spaces (TVSs) is called a bounded linear operator or just bounded if whenever
A subset of a TVS is called bounded (or more precisely, von Neumann bounded) if every neighborhood of the origin absorbs it.
In a normed space (and even in a seminormed space), a subset is von Neumann bounded if and only if it is norm bounded.
Hence, for normed spaces, the notion of a von Neumann bounded set is identical to the usual notion of a norm-bounded subset.
[1] This implies that every continuous linear operator between metrizable TVS is bounded.
However, in general, a bounded linear operator between two TVSs need not be continuous.
This formulation allows one to define bounded operators between general topological vector spaces as an operator which takes bounded sets to bounded sets.
In this context, it is still true that every continuous map is bounded, however the converse fails; a bounded operator need not be continuous.
This also means that boundedness is no longer equivalent to Lipschitz continuity in this context.
For LF spaces, a weaker converse holds; any bounded linear map from an LF space is sequentially continuous.
is a linear operator between two topological vector spaces and if there exists a neighborhood
[2] This fact is often summarized by saying that a linear operator that is bounded on some neighborhood of the origin is necessarily continuous.
In particular, any linear functional that is bounded on some neighborhood of the origin is continuous (even if its domain is not a normed space).
is a bornological space if and only if for every locally convex TVS
be a linear operator between topological vector spaces (not necessarily Hausdorff).
are locally convex then the following may be add to this list: if
is locally convex then the following may be added to this list: Let
that maps a polynomial to its derivative is not bounded.
The space of all bounded linear operators from