Some properties of the vector-valued Banach ideal space E(X) derived from those of E and X.

Wolfgang Hensgen

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Let $X$ be a Banach space. We give characterizations of when $ℱ (Y, X)$ is a $u$ -ideal in $𝒲 (Y, X)$ for every Banach space $Y$ in terms of nets of finite rank operators approximating weakly compact operators. Similar characterizations are given for the cases when $ℱ (X, Y)$ is a $u$ -ideal in $𝒲 (X, Y)$ for every Banach space $Y$ , when $ℱ (Y, X)$ is a $u$ -ideal in $𝒲 (Y, X^{* *})$ for every Banach space $Y$ , and when $ℱ (Y, X)$ is a $u$ -ideal in $𝒦 (Y, X^{* *})$ for every Banach space $Y$ .

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Given a vector measure m with values in a Banach space X, a desirable property (when available) of the associated Banach function space L¹(m) of all m-integrable functions is that L¹(m) = L¹(|m|), where |m| is the [0,∞]-valued variation measure of m. Closely connected to m is its X-valued integration map Iₘ: f ↦ ∫f dm for f ∈ L¹(m). Many traditional operators from analysis arise as integration maps in this way. A detailed study is made of the connection between the property L¹(m) = L¹(|m|)...