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A note on continuous restrictions of linear maps between Banach spaces. (English) Zbl 0823.47002

Summary: This note is devoted to the answers to the following questions asked by V. I. Bogachev, B. Kirchheim and W. Schachermayer [Acta Univ. Carol. Math. Phys. 30, No. 2, 31-35 (1989; Zbl 0715.47002)].
1. Let \(T: \ell_ 1\to X\) be a linear map into the infinite-dimensional Banach space \(X\). Can one find a closed infinite-dimensional subspace \(Z\subset \ell_ 1\) such that \(T\bigl|_ Z\) is continuous?
2. Let \(X= c_ 0\) or \(X= \ell_ p\) \((1< p< \infty)\) and let \(T: X\to X\) be a linear map. Can one find a dense subspace \(Z\) of \(X\) such that \(T\bigl|_ Z\) is continuous.

MSC:

47A05 General (adjoints, conjugates, products, inverses, domains, ranges, etc.)
46B20 Geometry and structure of normed linear spaces
46B45 Banach sequence spaces

Citations:

Zbl 0715.47002
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