arXiv subjects

Saurabh Dwivedi

Publications and source records attributed to Saurabh Dwivedi.

3 recordsLinked to original sources

arXiv2026

Weak$^*$-weak points of continuity on the state spaces

Let $X$ be a Banach space. For $x \in X$ with $\|x\| = 1$, we denote the state space by $S_x = \{x^* \in X^* : \|x^*\| = x^*(x) = 1\}.$ In this paper, we study weak$^*$-weak and weak$^*$-$\|\cdot\|$ points of continuity of the identity map on the state spaces in the space $\ell^p(X)$ for $1 < p < \infty$, where $X$ is a non-reflexive Banach space. We then use these results to characterize the weak and norm compactness of the state spaces of unit vectors in $\ell^p(X)$. In addition, we address an open problem concerning the characterization of weakly compact state spaces in the space of Bochner-integrable functions $L^1(μ, X)$. We also provide a local solution to this problem without any additional assumptions on the Banach space $X$. Motivated by the work of S. Daptari, V. Montesinos, and T. S. S. R. K. Rao, we show that if the set of all weak$^*$-weak points of continuity of $L^1(μ, X)_1^*$ is weakly dense in $L^1(μ, X)_1^*$, then $X^*$ has the Radon-Nikodým property (RNP).

math.FA↗

arXiv2026

An extension of Phelps theorem to spaces of vector-valued functions

In this paper, our main aim is to extend a classical theorem of Phelps on norm-attaining functionals from the space of scalar-valued continuous functions $C(Ω)$ to its vector-valued counterpart $C(Ω, X)$. One of our main results provides a complete characterization of norm-attaining functionals on $C(Ω, X)$ under the assumption that $X^*$ has the Radon-Nikodým property (RNP). For a general Banach space $X$, we further investigate norm attainment at points of weak$^*$-to-weak continuity for the identity map $Id : (C(Ω, X)_1^*, w^*) \to (C(Ω, X)_1^*, w)$.

math.FA↗

arXiv2025

A study on state spaces in classical Banach spaces

Let $X$ be a real or complex Banach space. Let $S(X)$ denote the unit sphere of $X$. For $x\in S(X)$, let $S_{x}=\{x^*\in S(X^*):x^*(x)=1\}$. A lot of Banach space geometry can be determined by the `quantum' of the state space $S_{x}$. In this paper, we mainly study the norm compactness and weak compactness of the state space in the space of Bochner integrable function and $c_{0}$-direct sums of Banach spaces. Suppose $X$ is such that $X^*$ is separable and let $μ$ be the Lebesgue measure on $[0,1]$. For $f\in L^1(μ,X)$, we demonstrate that if $S_{f}$ is norm compact, then $f$ is a smooth point. When $μ$ is the discrete measure, we show that if $ (x_i) \in S(\ell^{1}(X))$ and $ \|x_{i}\|\neq 0$ for all $i\in{\mathbb{N}}$, then $ S_{(x_i)}$ is weakly compact in $ \ell^\infty(X^*) $ if and only if $ S_{\frac{x_i}{\|x_i\|}} $ is weakly compact in $X^*$ for each $i\in{\mathbb{N}}$ and $\text{diam}\left(S_{\frac{x_i}{\|x_i\|}}\right) \to 0 $. For discrete $c_{0}$-sums, we show that for $(x_{i})\in c_{0}(X)$, $S_{(x_{i})}$ is weakly compact if and only if for each $i_{0}\in \mathbb{N}$ such that $\|x_{i_{0}}\|=1$, the state space $S_{x_{i_{0}}}$ is weakly compact.

math.FA↗