Polynomially reflexive space
Encyclopedia
In mathematics
, a polynomially reflexive space is a Banach space
X, on which the space of all polynomials in each degree is a reflexive space
.
Given a multilinear functional
Mn of degree n (that is, Mn is n-linear), we can define a polynomial p as
(that is, applying Mn on the diagonal
) or any finite sum of these. If only n-linear functionals are in the sum, the polynomial is said to be n-homogeneous.
We define the space Pn as consisting of all n-homogeneous polynomials.
The P1 is identical to the dual space
, and is thus reflexive for all reflexive X. This implies that reflexivity is a prerequisite for polynomial reflexivity.
x↦f(x) is always a (finite) linear combination of products x↦g(x) h(x) of two linear functional
s g and h. Therefore, assuming that the scalars are complex numbers, every sequence xn satisfying g(xn) → 0 for all linear functionals g, satisfies also f(xn) → 0 for all quadratic forms f.
In infinite dimension the situation is different. For example, in a Hilbert space
, an orthonormal sequence xn satisfies g(xn) → 0 for all linear functionals g, and nevertheless f(xn) = 1 where f is the quadratic form f(x) = ||x||2. In more technical words, this quadratic form fails to be weakly
sequentially continuous at the origin.
On a reflexive
Banach space
with the approximation property
the following two conditions are equivalent:
Quadratic forms are 2-homogeneous polynomials. The equivalence mentioned above holds also for n-homogeneous polynomials, n=3,4,...
spaces
, the Pn is reflexive if and only if n < p. Thus, no
is polynomially reflexive. (
is ruled out because it is not reflexive.)
Thus if space contains
as a quotient space
, it is not polynomially reflexive. This makes polynomially reflexive spaces rare.
The symmetric Tsirelson space
is polynomially reflexive.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, a polynomially reflexive space is a Banach space
Banach space
In mathematics, Banach spaces is the name for complete normed vector spaces, one of the central objects of study in functional analysis. A complete normed vector space is a vector space V with a norm ||·|| such that every Cauchy sequence in V has a limit in V In mathematics, Banach spaces is the...
X, on which the space of all polynomials in each degree is a reflexive space
Reflexive space
In functional analysis, a Banach space is called reflexive if it coincides with the dual of its dual space in the topological and algebraic senses. Reflexive Banach spaces are often characterized by their geometric properties.- Normed spaces :Suppose X is a normed vector space over R or C...
.
Given a multilinear functional
Functional (mathematics)
In mathematics, and particularly in functional analysis, a functional is a map from a vector space into its underlying scalar field. In other words, it is a function that takes a vector as its input argument, and returns a scalar...
Mn of degree n (that is, Mn is n-linear), we can define a polynomial p as
(that is, applying Mn on the diagonal
Diagonal
A diagonal is a line joining two nonconsecutive vertices of a polygon or polyhedron. Informally, any sloping line is called diagonal. The word "diagonal" derives from the Greek διαγώνιος , from dia- and gonia ; it was used by both Strabo and Euclid to refer to a line connecting two vertices of a...
) or any finite sum of these. If only n-linear functionals are in the sum, the polynomial is said to be n-homogeneous.
We define the space Pn as consisting of all n-homogeneous polynomials.
The P1 is identical to the dual space
Dual space
In mathematics, any vector space, V, has a corresponding dual vector space consisting of all linear functionals on V. Dual vector spaces defined on finite-dimensional vector spaces can be used for defining tensors which are studied in tensor algebra...
, and is thus reflexive for all reflexive X. This implies that reflexivity is a prerequisite for polynomial reflexivity.
Relation to continuity of forms
On a finite-dimensional linear space, a quadratic formQuadratic form
In mathematics, a quadratic form is a homogeneous polynomial of degree two in a number of variables. For example,4x^2 + 2xy - 3y^2\,\!is a quadratic form in the variables x and y....
x↦f(x) is always a (finite) linear combination of products x↦g(x) h(x) of two linear functional
Linear functional
In linear algebra, a linear functional or linear form is a linear map from a vector space to its field of scalars. In Rn, if vectors are represented as column vectors, then linear functionals are represented as row vectors, and their action on vectors is given by the dot product, or the...
s g and h. Therefore, assuming that the scalars are complex numbers, every sequence xn satisfying g(xn) → 0 for all linear functionals g, satisfies also f(xn) → 0 for all quadratic forms f.
In infinite dimension the situation is different. For example, in a Hilbert space
Hilbert space
The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra and calculus from the two-dimensional Euclidean plane and three-dimensional space to spaces with any finite or infinite number of dimensions...
, an orthonormal sequence xn satisfies g(xn) → 0 for all linear functionals g, and nevertheless f(xn) = 1 where f is the quadratic form f(x) = ||x||2. In more technical words, this quadratic form fails to be weakly
Weak convergence (Hilbert space)
In mathematics, weak convergence in a Hilbert space is convergence of a sequence of points in the weak topology.-Definition:A sequence of points in a Hilbert space H is said to converge weakly to a point x in H if...
sequentially continuous at the origin.
On a reflexive
Reflexive space
In functional analysis, a Banach space is called reflexive if it coincides with the dual of its dual space in the topological and algebraic senses. Reflexive Banach spaces are often characterized by their geometric properties.- Normed spaces :Suppose X is a normed vector space over R or C...
Banach space
Banach space
In mathematics, Banach spaces is the name for complete normed vector spaces, one of the central objects of study in functional analysis. A complete normed vector space is a vector space V with a norm ||·|| such that every Cauchy sequence in V has a limit in V In mathematics, Banach spaces is the...
with the approximation property
Approximation property
In mathematics, a Banach space is said to have the approximation property , if every compact operator is a limit of finite-rank operators. The converse is always true....
the following two conditions are equivalent:
- every quadratic form is weakly sequentially continuous at the origin;
- the Banach space of all quadratic forms is reflexive.
Quadratic forms are 2-homogeneous polynomials. The equivalence mentioned above holds also for n-homogeneous polynomials, n=3,4,...
Examples
For the spacesLp space
In mathematics, the Lp spaces are function spaces defined using a natural generalization of the p-norm for finite-dimensional vector spaces...
, the Pn is reflexive if and only if n < p. Thus, no
is polynomially reflexive. ( is ruled out because it is not reflexive.)Thus if space contains
as a quotient spaceQuotient space (linear algebra)
In linear algebra, the quotient of a vector space V by a subspace N is a vector space obtained by "collapsing" N to zero. The space obtained is called a quotient space and is denoted V/N ....
, it is not polynomially reflexive. This makes polynomially reflexive spaces rare.
The symmetric Tsirelson space
Tsirelson space
In mathematics, Tsirelson space T is an example of a reflexive Banach space in which neither an l p space nor a c0 space can be embedded.It was introduced by B. S. Tsirelson in 1974...
is polynomially reflexive.