This is a side question from [this](https://mathoverflow.net/questions/420655/infinite-dimensional-algebraic-varieties), and I have also asked it on [MSE](https://math.stackexchange.com/questions/4434331/an-explicit-description-for-homogeneous-polynomials-in-lp-space) without getting any respond.

Denote $l^p (1 \le p \le \infty)$ as the complex Banach space of complex sequences with finite $p$-norm and $c_0$ as the closed subspace of $l^{\infty}$ containing complex sequences with limit $0$. Suppose $(x_i):=(x_0, x_1, \cdots) \in l^p/c_0$, and define the "degree-$d$ Veronese map" as $$V_d^p: l^p \rightarrow l^p: (x_i) \mapsto \left( \left( \frac{d!}{l_{1}! \cdots l_{d}!} \right)^{\frac{1}{p}} x_{k_1}^{l_1} \cdots x_{k_d}^{l_d} \right)(\frac{1}{\infty}:=0)$$ where $0 \le l_1 \le \cdots \le l_d \le d$ values in every non-negative integer partition of $d$ and the coordinates of the image are arranged with the increase in $l_1 k_1 + \cdots + l_d k_d$. It can be shown that $V_d^p$ is well-defined (including mapping $c_0$ to $c_0$) and continuous since $\lVert V_d^p(x) \rVert_p = \lVert x \rVert_p^d$.

Suppose $A$ is a symmetric $d$-linear functional on $X = l^p/c_0$. Define the degree-$d$ homogeneous polynomial w.r.t $A$ as $p_A(x)=A(\underbrace{x, \cdots ,x}_{d \text{ times}})$, 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 $L:X^*=P^1(X) \rightarrow P^d(X):\psi \mapsto \psi \circ V_d^p$ is a norm-$1$ injective linear map. **How to describe the image of this map as a subspace of $P^d(X)$?** In particular, **does this subspace precisely consist of all weakly sequentially continuous (i.e. the image of a weakly-convergent sequence is norm-convergent) 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.

**PS**: I welcome answers of a full explicit description of all homogeneous polynomials on $l^p/c_0$, since the most "canonical" polynomials $x_0^d+x_1^d+ \cdots$ for $d \ge p$ seems to have no weak-topology-based continuity.