New proofs of Rosenthal's $\ell^{1}$--theorem and the Josefson--Nissenzweig theorem

The Josefson--Nissenzweig theorem, Grothendieck property, and finitely supported measures on compact spaces

The celebrated Josefson-Nissenzweig theorem implies that for a Banach space $C(K)$ of continuous real-valued functions on an infinite compact space $K$ there exists a sequence of Radon measures $\langleμ_n\colon\ n\inω\rangle$ on $K$ which is weakly* convergent to the zero measure on $K$ and such that $\big\|μ_n\big\|=1$ for every $n\inω$. We call such a sequence of measures \textit{a Josefson-Nissenzweig sequence}. In this paper we study the situation when the space $K$ admits a Josefson-Nissenzweig sequence of measures such that its every element has finite support. We prove among the others…

The Josefson--Nissenzweig property for locally convex spaces

We define a locally convex space $E$ to have the $Josefson$-$Nissenzweig$ $property$ (JNP) if the identity map $(E',σ(E',E))\to ( E',β^\ast(E',E))$ is not sequentially continuous. By the classical Josefson-Nissenzweig theorem, every infinite-dimensional Banach space has the JNP. A characterization of locally convex spaces with the JNP is given. We thoroughly study the JNP in various function spaces. Among other results we show that for a Tychonoff space $X$, the function space $C_p(X)$ has the JNP iff there is a weak$^\ast$ null-sequence $(μ_n)_{n\inω}$ of finitely supported sign-measures on $…

Josefson-Nissenzweig property for $C_p$-spaces

The famous Rosenthal-Lacey theorem asserts that for each infinite compact space $K$ the Banach space $C(K)$ admits a quotient which is either a copy of $c_{0}$ or $\ell_{2}$. The aim of the paper is to study a natural variant of this result for the space $C_{p}(K)$ of continuous real-valued maps on $K$ with the pointwise topology. Following famous Josefson-Nissenzweig theorem for infinite-dimensional Banach spaces we introduce a corresponding property (called Josefson-Nissenzweig property, briefly, the JNP) for $C_{p}$-spaces. We prove: For a Tychonoff space $X$ the space $C_p(X)$ satisfies th…

Banach空间C(K×L)的Josefson-Nissenzweig定理的初等证明

An Elementary Proof Of The Josefson-Nissenzweig Theorem For Banach Spaces C(KxL)

本文通过初等组合算术方法，给出了Banach空间C(K×L)的Josefson-Nissenzweig定理的证明，并推广了Cembranos和Freniche的经典定理。

Grothendieck $C(K)$-空间与Josefson--Nissenzweig定理

Grothendieck $C(K)$-spaces and the Josefson--Nissenzweig theorem

本文研究紧空间上连续函数空间的 $\ell_1$-Grothendieck 性质，证明其等价于不存在满足 Josefson--Nissenzweig 定理的序列，构造了可分紧空间使该性质成立但标准 Grothendieck 性质不成立，并指出经典 Efimov 空间缺乏该性质。

The Josefson--Nissenzweig theorem and filters on $ω$

For a free filter $F$ on $ω$, endow the space $N_F=ω\cup\{p_F\}$, where $p_F\not\inω$, with the topology in which every element of $ω$ is isolated whereas all open neighborhoods of $p_F$ are of the form $A\cup\{p_F\}$ for $A\in F$. Spaces of the form $N_F$ constitute the class of the simplest non-discrete Tychonoff spaces. The aim of this paper is to study them in the context of the celebrated Josefson--Nissenzweig theorem from Banach space theory. We prove, e.g., that, for a filter $F$, the space $N_F$ carries a sequence $\langleμ_n\colon n\inω\rangle$ of normalized finitely supported signed …