Relation To Continuity of Forms
On a finite-dimensional linear space, a quadratic form x↦f(x) is always a (finite) linear combination of products x↦g(x) h(x) of two linear functionals 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:
- 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,...
Read more about this topic: Polynomially Reflexive Space
Famous quotes containing the words relation to, relation, continuity and/or forms:
“The proper study of mankind is man in his relation to his deity.”
—D.H. (David Herbert)
“You know there are no secrets in America. It’s quite different in England, where people think of a secret as a shared relation between two people.”
—W.H. (Wystan Hugh)
“Every society consists of men in the process of developing from children into parents. To assure continuity of tradition, society must early prepare for parenthood in its children; and it must take care of the unavoidable remnants of infantility in its adults. This is a large order, especially since a society needs many beings who can follow, a few who can lead, and some who can do both, alternately or in different areas of life.”
—Erik H. Erikson (1904–1994)
“Culture’s essential service to a religion is to destroy intellectual idolatry, the recurrent tendency in religion to replace the object of its worship with its present understanding and forms of approach to that object.”
—Northrop Frye (b. 1912)