关于S.Dayal论文中Banach空间局部表示定理的定理3.2证明疑问

Hey there, let's unpack this question about Theorem 3.2 in S. Dayal's Local Representation Theorems on Banach Spaces—I’ve spent some time working with these kinds of polynomial Banach space results, so let’s break this down.

First, let's anchor this in the standard framework of Banach space polynomials: any polynomial (P: X \to Y) (where (X,Y) are Banach spaces) has a unique decomposition into homogeneous polynomials:
(P(x) = \sum_{k=0}^n P_k(x))
where (P_k) is the homogeneous polynomial of degree (k)—this is exactly the "coefficient of (t^k)" you’re referring to when expanding (P(x + ty)) or similar (a common technique in the paper).

Here’s why the theorem’s conclusion for (P) applies directly to each (P_k):

• Uniqueness of Polynomial Decomposition: The key point here is that this homogeneous decomposition is unique. If (P) satisfies the local representation property from Theorem 3.2, then each (P_k) must as well. Think of it this way: if the entire sum has a local factorization or approximation, you can’t have one component violating that property without breaking the result for the full polynomial.
• Polarization and Multilinear Links: Most local representation theorems for polynomials rely on connecting homogeneous polynomials to multilinear maps via polarization formulas. If the theorem’s proof uses this link for (P), the polarization process will naturally isolate each (P_k) and carry over the representation property to it. For example, if (P) has a local representation as a composition of linear maps, each (P_k) will inherit a corresponding multilinear representation.
• Inherited Continuity/Boundedness: Banach space polynomials are continuous if and only if every homogeneous component is continuous. Theorem 3.2 almost certainly uses continuity or boundedness of (P) as a premise, so those properties directly transfer to each (P_k)—a necessary condition for the theorem’s conclusion to hold for the coefficients.
• Direct Sum Structure: The space of polynomials on (X) to (Y) is a direct sum of the spaces of homogeneous polynomials of each degree. Many functional analysis theorems that hold for direct sums automatically hold for each summand, which is exactly what’s happening here: the local representation property is preserved across this direct sum, so each (P_k) (your (t^k) coefficient) gets the same result.

If you’re stuck on a specific line in the proof—say, a step where the author jumps from (P) to its coefficients without explanation—feel free to share that exact snippet, and we can walk through the logical gap in more detail!

内容的提问来源于stack exchange，提问作者william paine