Continuous linear extension
Encyclopedia
In functional analysis
, it is often convenient to define a linear transformation
on a complete
, normed vector space
by first defining a linear transformation 
on a dense
subset
of
and then extending 
to the whole space via the theorem below. The resulting extension remains linear and bounded
(thus continuous
).
This procedure is known as continuous linear extension.
from a normed vector space 
to a complete, normed vector space 
can be uniquely extended to a bounded linear transformation 
from the completion of 
to 
. In addition, the operator norm
of
is 
iff
the norm of
is 
.
This theorem is sometimes called the B L T theorem, where B L T stands for bounded linear transformation.
. A step function
on a closed
interval
is a function of the form: 
where
are real numbers, 
, and 
denotes the indicator function of the set 
. The space of all step functions on 
, normed by the 
norm (see Lp space
), is a normed vector space which we denote by
. Define the integral of a step function by: 
.
as a function is a bounded linear transformation from 
into 
.
Let
denote the space of bounded, piecewise
continuous functions on
that are continuous from the right, along with the 
norm. The space 
is dense in 
, so we can apply the B.L.T. theorem to extend the linear transformation 
to a bounded linear transformation 
from 
to 
. This defines the Riemann integral of all functions in 
; for every 
, 
.
to a bounded linear transformation from 
to 
, if 
is dense in 
. If 
is not dense in 
, then the Hahn–Banach theorem
may sometimes be used to show that an extension exists
. However, the extension may not be unique.
Functional analysis
Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure and the linear operators acting upon these spaces and respecting these structures in a suitable sense...
, it is often convenient to define a linear transformation
Linear transformation
In mathematics, a linear map, linear mapping, linear transformation, or linear operator is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication. As a result, it always maps straight lines to straight lines or 0...
on a complete
Complete space
In mathematical analysis, a metric space M is called complete if every Cauchy sequence of points in M has a limit that is also in M or, alternatively, if every Cauchy sequence in M converges in M....
, normed vector space
Normed vector space
In mathematics, with 2- or 3-dimensional vectors with real-valued entries, the idea of the "length" of a vector is intuitive and can easily be extended to any real vector space Rn. The following properties of "vector length" are crucial....
by first defining a linear transformation on a denseDense set
In topology and related areas of mathematics, a subset A of a topological space X is called dense if any point x in X belongs to A or is a limit point of A...
subset
Subset
In mathematics, especially in set theory, a set A is a subset of a set B if A is "contained" inside B. A and B may coincide. The relationship of one set being a subset of another is called inclusion or sometimes containment...
of
and then extending to the whole space via the theorem below. The resulting extension remains linear and boundedBounded 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 to that of v is bounded by the same number, over all non-zero vectors v in X...
(thus continuous
Continuous function
In mathematics, a continuous function is a function for which, intuitively, "small" changes in the input result in "small" changes in the output. Otherwise, a function is said to be "discontinuous". A continuous function with a continuous inverse function is called "bicontinuous".Continuity of...
).
This procedure is known as continuous linear extension.
Theorem
Every bounded linear transformation from a normed vector space to a complete, normed vector space can be uniquely extended to a bounded linear transformation from the completion of to . In addition, the operator normOperator norm
In mathematics, the operator norm is a means to measure the "size" of certain linear operators. Formally, it is a norm defined on the space of bounded linear operators between two given normed vector spaces.- Introduction and definition :...
of
is iffIFF
IFF, Iff or iff may refer to:Technology/Science:* Identification friend or foe, an electronic radio-based identification system using transponders...
the norm of
is .This theorem is sometimes called the B L T theorem, where B L T stands for bounded linear transformation.
Application
Consider, for instance, the definition of the Riemann integralRiemann integral
In the branch of mathematics known as real analysis, the Riemann integral, created by Bernhard Riemann, was the first rigorous definition of the integral of a function on an interval. The Riemann integral is unsuitable for many theoretical purposes...
. A step function
Step function
In mathematics, a function on the real numbers is called a step function if it can be written as a finite linear combination of indicator functions of intervals...
on a closed
Closure (mathematics)
In mathematics, a set is said to be closed under some operation if performance of that operation on members of the set always produces a unique member of the same set. For example, the real numbers are closed under subtraction, but the natural numbers are not: 3 and 8 are both natural numbers, but...
interval
Interval (mathematics)
In mathematics, a interval is a set of real numbers with the property that any number that lies between two numbers in the set is also included in the set. For example, the set of all numbers satisfying is an interval which contains and , as well as all numbers between them...
is a function of the form: where
are real numbers, , and denotes the indicator function of the set . The space of all step functions on , normed by the norm (see Lp spaceLp space
In mathematics, the Lp spaces are function spaces defined using a natural generalization of the p-norm for finite-dimensional vector spaces...
), is a normed vector space which we denote by
. Define the integral of a step function by: . as a function is a bounded linear transformation from into .Let
denote the space of bounded, piecewisePiecewise
On mathematics, a piecewise-defined function is a function whose definition changes depending on the value of the independent variable...
continuous functions on
that are continuous from the right, along with the norm. The space is dense in , so we can apply the B.L.T. theorem to extend the linear transformation to a bounded linear transformation from to . This defines the Riemann integral of all functions in ; for every , .The Hahn–Banach theorem
The above theorem can be used to extend a bounded linear transformation to a bounded linear transformation from to , if is dense in . If is not dense in , then the Hahn–Banach theoremHahn–Banach theorem
In mathematics, the Hahn–Banach theorem is a central tool in functional analysis. It allows the extension of bounded linear functionals defined on a subspace of some vector space to the whole space, and it also shows that there are "enough" continuous linear functionals defined on every normed...
may sometimes be used to show that an extension exists
Existence
In common usage, existence is the world we are aware of through our senses, and that persists independently without them. In academic philosophy the word has a more specialized meaning, being contrasted with essence, which specifies different forms of existence as well as different identity...
. However, the extension may not be unique.