This is a side question from Infinite-dimensional "algebraic varieties", and I have also asked it on MSE without getting any response.

Denote by $l^p$ ($1 \le p \le \infty$) as the complex Banach space of complex sequences with finite $p$-norm and by $c_0$ the closed subspace of $l^{\infty}$ containing complex sequences with limit $0$. Suppose $(x_i):=(x_0, x_1, \dotsc)$, then the "degree-$d$ Veronese map" can be defined as $$V_d^p: l^p \rightarrow l^p: (x_i) \mapsto \left( \left( \frac{d!}{l_{1}! \dotsm l_{d}!} \right)^{\frac{1}{p}} x_{k_1}^{l_1} \dotsm x_{k_d}^{l_d} \right)\quad\left(\frac{1}{\infty}:=0\right)$$ where $0 \le l_1 \le \dotsb \le l_d \le d$ runs over non-negative integer partitions of $d$ and the coordinates of the image are arranged in increasing order of $l_1 k_1 + \dotsb + l_d k_d$. It can be shown that $V_d^p$ is well-defined and continuous since $\lVert V_d^p(x) \rVert_p = \lVert x \rVert_p^d$. Moreover, $V_d^{\infty}(c_0) \subseteq c_0$.

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 $p_A(x)=A(\underbrace{x, \dotsc ,x}_{d \text{ times}})$, 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 $L:X^*=P^1(X) \rightarrow P^d(X):\psi \mapsto \psi \circ V_d^p$ 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 $x_0^d+x_1^d+ \dotsb$ for $d \ge p$ seem to have no weak-topology-based continuity.