In functional analysis, a bounded linear operator is a linear transformation between topological vector spaces (TVSs) and that maps bounded subsets of to bounded subsets of If and are normed vector spaces (a special type of TVS), then is bounded if and only if there exists some such that for all in
The smallest such denoted by is called the operator norm of
A linear operator that is sequentially continuous or continuous is a bounded operator and moreover, a linear operator between normed spaces is bounded if and only if it is continuous. However, a bounded linear operator between more general topological vector spaces is not necessarily continuous.
A linear operator between two topological vector spaces (TVSs) is locally bounded or just bounded if whenever is bounded in then is bounded in 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.
Every sequentially continuous linear operator between TVS is a bounded operator.This implies that every continuous linear operator 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. Clearly, this also means that boundedness is no longer equivalent to Lipschitz continuity in this context.
If the domain is a bornological space (e.g. a pseudometrizable TVS, a Fréchet space, a normed space) then a linear operators into any other locally convex spaces is bounded if and only if it is continuous. For LF spaces, a weaker converse holds; any bounded linear map from an LF space is sequentially continuous.
Bornological spaces are exactly those locally convex spaces for every bounded linear operator into another locally convex space is necessarily continuous. That is, a locally convex TVS is a bornological space if and only if for every locally convex TVS a linear operator is continuous if and only if it is bounded.
Every normed space is bornological.
Let be a linear operator between TVSs (not necessarily Hausdorff). The following are equivalent:
and if in addition and are locally convex then the following may be add to this list:
and if in addition is a bornological space and is locally convex then the following may be added to this list:
A bounded linear operator is generally not a bounded function, as generally one can find a sequence in such that Instead, all that is required for the operator to be bounded is that
for all So, the operator could only be a bounded function if it satisfied for all as is easy to understand by considering that for a linear operator, for all scalars Rather, a bounded linear operator is a locally bounded function.
A linear operator between normed spaces is bounded if and only if it is continuous, and by linearity, if and only if it is continuous at zero.
A linear operator between normed spaces and is bounded if and only if it is a continuous linear operator. A proof is given below.
Proof of equivalence of boundedness and continuity
Suppose that is bounded. Then, for all vectors with nonzero we have
Conversely, it follows from the continuity at the zero vector that there exists a such that for all vectors with Thus, for all non-zero one has
This proves that is bounded.
The condition for to be bounded, namely that there exists some such that for all
is precisely the condition for to be Lipschitz continuous at 0 (and hence everywhere, because is linear).
A common procedure for defining a bounded linear operator between two given Banach spaces is as follows. First, define a linear operator on a dense subset of its domain, such that it is locally bounded. Then, extend the operator by continuity to a continuous linear operator on the whole domain.
is a continuous function, then the operator defined on the space of continuous functions on endowed with the uniform norm and with values in the space with given by the formula
Not every linear operator between normed spaces is bounded. Let be the space of all trigonometric polynomials defined on with the norm
Define the operator which acts by taking the derivative, so it maps a polynomial to its derivative Then, for
with we have while as so this operator is not bounded.
It turns out that this is not a singular example, but rather part of a general rule. However, given any normed spaces and with infinite-dimensional and not being the zero space, one can find a linear operator which is not continuous from to
That such a basic operator as the derivative (and others) is not bounded makes it harder to study. If, however, one defines carefully the domain and range of the derivative operator, one may show that it is a closed operator. Closed operators are more general than bounded operators but still "well-behaved" in many ways.
In mathematics, more specifically in functional analysis, a Banach space is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vectors and is complete in the sense that a Cauchy sequence of vectors always converges to a well defined limit that is within the space.
In mathematics, any vector space has a corresponding dual vector space consisting of all linear forms on , together with the vector space structure of pointwise addition and scalar multiplication by constants.
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In functional analysis and related areas of mathematics, Fréchet spaces, named after Maurice Fréchet, are special topological vector spaces. They are generalizations of Banach spaces. All Banach and Hilbert spaces are Fréchet spaces. Spaces of infinitely differentiable functions are typical examples of Fréchet spaces, many of which are typically not Banach spaces.
In mathematics, the uniform boundedness principle or Banach–Steinhaus theorem is one of the fundamental results in functional analysis. Together with the Hahn–Banach theorem and the open mapping theorem, it is considered one of the cornerstones of the field. In its basic form, it asserts that for a family of continuous linear operators whose domain is a Banach space, pointwise boundedness is equivalent to uniform boundedness in operator norm.
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In mathematics, a nuclear space is a topological vector space with many of the good properties of finite-dimensional vector spaces. The topology on them can be defined by a family of seminorms whose unit balls decrease rapidly in size. Vector spaces whose elements are "smooth" in some sense tend to be nuclear spaces; a typical example of a nuclear space is the set of smooth functions on a compact manifold.
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This is a glossary for the terminology in a mathematical field of functional analysis.
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