# An explicit description for a certain type of infinite-dimensional homogeneous polynomials

This is a side question from Infinite-dimensional "algebraic varieties".

Denote by $$X_p$$ ($$1 \le p \le \infty$$) the Banach spaces of complex sequences with finite $$p$$-norm and limit $$0$$. Suppose $$(x_i):=(x_0, x_1, \dotsc)$$, then the "degree-$$d$$ Veronese map" can be defined as $$V_p^d: X_p \rightarrow X_p: (x_i) \mapsto \left( \left( \frac{d!}{l_{1}! \dotsb l_{d}!} \right)^{\frac{1}{p}} x_{k_1}^{l_1} \dotsb x_{k_d}^{l_d} \right)\quad\left(\frac{1}{\infty}:=0\right)$$ where $$l_1, \dotsb, l_d$$ runs over non-negative integer partitions of $$d$$, $$0 \le k_1 \lt \dotsb \lt k_d$$ and the coordinates of the right side are arranged in increasing order of $$l_1 k_1 + \dotsb + l_d k_d$$. It can be shown that $$V_p^d$$ is well-defined and continuous since $$\lVert V_p^d(x) \rVert_p = \lVert x \rVert_p^d$$.

Suppose $$A$$ is a symmetric $$d$$-linear functional on some $$X=X_p$$. Define the degree-$$d$$ homogeneous polynomial w.r.t. $$A$$ as $$P_A(x)=A(\underbrace{x, \dotsc ,x}_{d \text{ times}})$$, and equip the vector space of degree-$$d$$ homogeneous polynomials on $$X$$, $$P^d(X)$$, with the norm $$\lVert P \rVert = \sup_{\lVert x \rVert = 1}\lvert P(x) \rvert$$ to make it a Banach space. It can be seen that $$L:X^*=P^1(X) \rightarrow P^d(X):\psi \mapsto \psi \circ V_p^d$$ is a norm-$$1$$ injective linear map. How to describe $$\operatorname{Ran}(L)$$ as a subspace of $$P^d(X)$$? In particular, does this subspace precisely consist of all completely continuous (i.e. mapping weakly-convergent sequences to norm-convergent sequences) polynomials? If not, I want to know the strongest continuity (e.g. weakly-continuous on every bounded set) that the polynomials in this subspace could reach.

Similar statements can also be made for $$X'_\infty=l^\infty$$. However, since $$l^\infty$$ is inseperable and has a complicated dual under $$ZFC$$, I'd rather exclude it for a bonus question.

PS: I welcome answers of a full explicit description of all homogeneous polynomials on $$X_p$$, since the most "canonical" homogeneous polynomials $$x_0^d+x_1^d+ \dotsb$$ for $$(x_i) \in X_p,d \ge p$$ seem to have no weak-topology-based continuity.

Progress $$1$$: Until now the only solved case is $$p=1$$. We only give a proof of the case $$d=2$$ (the other values are similar): Suppose $$P \in P^2(X_1)$$. Denote $$P_{ij}$$ by the Gateaux derivative of $$P$$ along $$e_i,e_j$$, where $$e_i \in X_1$$ is the $$i$$-th unit vector. It's not hard to show that $$P_{ij} \equiv 0 \implies P \equiv 0$$ by the continuity of $$P$$ (Suppose on the contrary that $$P \ne 0$$, then $$\exists v \in X_1$$ such that $$P(v) \ne 0$$. Examine the values of $$P$$ on the tails of $$v$$ to get a contradiction.), thus $$P$$ can be formally written as $$P((x_i))=\sum_{i=0}^{+\infty}a_ix_i^2 + 2\sum_{0 \le i \lt j}b_{ij}x_ix_j$$ Then it is sufficient to prove that $$\{ a_i \} \cup \{ b_{ij} \}$$ is bounded. If $$\{ a_i \}$$ is unbounded, then $$P(e_i)=a_i$$ is unbounded; if $$\{ a_i \}$$ is bounded but $$\{ b_{ij} \}$$ is unbounded, then $$P(e_i+e_j)=2b_{ij}+(a_i+a_j)$$ $$(i \ne j)$$ is the sum of an unbounded sequence with a bounded sequence, which is unbounded (For $$d \gt 2$$ the coefficients in front of $$e_j$$ might be other numbers, e.g. in the case $$d=3$$, $$P(e_i-e_j)$$ is also involved.). Either case contradicts the continuity of $$P$$. We can see that $$L$$ is actually an isomorphism between $$P^1(X_1)$$ and $$P^d(X_1)$$, which gives an affirmative answer for the case $$p=1$$ considering the Dunford-Pettis property of $$X_1$$. (If my proof is correct, it would be surprising that this non-trivial isomorphism doesn't occur anywhere else. So I'm wondering if there exist some related references.)

Progress $$2$$: It seems that there are still many completely continuous polynomials outside $$\operatorname{Ran}(L)$$ when $$d \ge p$$. For example, any $$Q(x)=a_0 x_0^2+a_1 x_1^2+\dotsb$$ with $$a_n \rightarrow 0$$ is a completely continuous polynomial on $$X_2$$, but $$Q \in \operatorname{Ran}(L) \iff (a_i) \in X_2$$ (just as the difference between compact operators and Hilbert-Schmidt operators). It is then more meaningful to investigate the closure of $$\operatorname{Ran}(L)$$ (under the norm topology).

• What is $l^p/c_0$ for $p<\infty$? Apr 27, 2022 at 13:24
• @M.González The backslash means "or", not quotient space. Apr 27, 2022 at 13:38
• Although it's clear what you meant in your comment, I mention that $/$ is a slash, not a backslash. But it's not so clear in the body, so I would encourage you just to say "or"—there's no character limit! Also, you might want to consider denoting your polynomials by something other than $p$ when discussing $l^p$ spaces …. Apr 27, 2022 at 16:49
• @LSpice Thanks for your advice. I've changed some formulation. Apr 27, 2022 at 16:56

$$\ell^1$$ and $$c_0$$ have the Dunford-Pettis property. Thus, every polynomial on these spaces are weakly sequentially continuous (w.s.c.), see Proposition 2.34 in the book by Dineen.
$$X=\ell^p$$ is reflexive for $$1. Thus, every w.s.c. bounded bilinear map is of the form $$A(x,y) = \langle Tx,y\rangle$$ for some compact operator $$T:X\to X^*$$. In particular, when $$p>2$$, every bounded linear $$T:\ell^p\to\ell^q$$ is compact by Pitt's theorem; thus, every bilinear map and every degree 2 polynomial is w.s.c.
• I remember I leant from somewhere else that degree-$d$ polynomials in $l^p$ for $p>d$ is weakly sequentially continuous. Is this statement correct? Apr 29, 2022 at 17:45