AI 中文总结
该研究针对p-凸Banach序列格上的2-齐次多项式与双线性型建立集中定理，推导其Kadec-Klee性质，明确最优条件并拓展至算子空间，还确立了相关张量范数的强次可微性。
AI 中文摘要
我们通过给出最优结果研究齐次多项式空间与多重线性型空间中的Kadec-Klee性质。实际上，我们的主要工具是一族集中定理，该定理表明，在合适的p-凸Banach序列格上，若一个2-齐次多项式或双线性型在有限支撑向量上几乎达到其范数，则它与该多项式或双线性型在对应有限坐标集上的限制是一致接近的。由此，我们得到复2-齐次多项式空间与复双线性型空间的弱星一致Kadec-Klee性质，这些空间是对p>2时的p-凸Banach序列格，且常数为1。这一结果尤其适用于c₀、ℓₚ、Lorentz空间d(w,p)、Garling空间g(w,p)等。在实情形中，基于幂型凸性模的集中论证为算子空间与双线性型空间带来进一步的正结果，拓展了已有文献中的结论。我们还证明，对于ℓₚ、d(w,p)和g(w,p)上的2-齐次多项式，p>2的限制是最优的，并确立了至少2次的实齐次多项式以及至少3次的复齐次多项式与对称多重线性型的序列弱星Kadec-Klee性质的普遍失效情况。这些结果表明，我们的正结果在次数方面是最优的，且假设p>2对于经典空间ℓₚ、d(w,p)和g(w,p)是最优的。作为我们结果的推论，我们确立了相关的射影与对称射影张量范数的强次可微性。
英文摘要
We study Kadec-Klee properties in spaces of homogeneous polynomials and multilinear forms by presenting optimal results. Indeed, our main tool is a family of concentration theorems showing that, on suitable $p$-convex Banach sequence lattices, a $2$-homogeneous polynomial or a bilinear form which almost attains its norm at finitely supported vectors is uniformly close to its restriction to the corresponding finite set of coordinates. As a consequence, we obtain the weak-star uniform Kadec-Klee property for spaces of complex $2$-homogeneous polynomials and complex bilinear forms on $p$-convex Banach sequence lattices with constant one for some $p>2$. This applies, in particular, to $c_0$, $\ell_p$, Lorentz spaces $d(w,p)$, Garling spaces $g(w,p)$, among others. In the real setting, a concentration argument based on the modulus of convexity of power type yields further positive results for spaces of operators and bilinear forms, extending previous results in the literature. We also prove that the restriction $p>2$ is optimal for $2$-homogeneous polynomials on $\ell_p$, $d(w,p)$ and $g(w,p)$, and establish general failures of the sequential weak-star Kadec-Klee property for real homogeneous polynomials of degree at least two and for complex homogeneous polynomials and symmetric multilinear forms of degree at least three. These results show that our positive results are optimal respect to the degree and that the assumption $p>2$ is optimal for the classical spaces $\ell_p$, $d(w,p)$ and $g(w,p)$. As a consequence of our results, we establish the strong subdifferentiability of the associated projective and symmetric projective tensor norms.
