What is the polynomial functions on the open unit ball of a Banach space?
In functional analysis, a polynomial function on the open unit ball of a Banach space refers to a function that can be expressed as a finite sum of powers of a given norm. More formally, let X be a Banach space, and let B(X) denote the open unit ball in X. A polynomial function on B(X) is a function f: B(X) -> X that can be written as:
f(x) = ∑_{n=0}^N a_n x^n
where a_n ∈ X are coefficients, x^n represents the n-th power of x in X (i.e., x multiplied by itself n times), and the sum is taken over a finite number of terms N.
The coefficients a_n can be chosen to ensure that the series converges in X for all x in B(X). The convergence is usually understood in terms of the norm topology on X.
Let X and Y be Banach spaces, n a positive integer.
A function P from X into Y is a n-homogeneous polynomial if there exists a
n-linear function L from X^n to Y such that P(x)=L(x,...,x).
A 0-homogeneous polynomial is a constant function.
A polynomial is the sum of a finite number of homogeneous polynomials.
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