• Journal of applied mathematics & informatics
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• v.17 no.1_2_3
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• pp.567-572
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• 2005

A Banach space X is a Hilbert space if and only if $$E{\mid}x=Y{\mid}{\le}2$$ for all x$\in$ X and all X-valued Bochner integrable functions Y satisfying EY = 0 and ${\mid}x-Y{\mid}{\le}2$ a.e.

• Journal of applied mathematics & informatics
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• v.16 no.1_2
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• pp.533-537
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• 2004

A Banach space X is a Hilbert space if and only if E｜x + Y｜$geq$ 1 for all x $\epsilon$ X and all X-valued Bochner integrable functions Y satisfying EY = 0 and ｜Y｜$geq$ 1 a.e.

• Communications of the Korean Mathematical Society
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• v.9 no.3
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• pp.599-606
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• 1994

In [1], Cameron and Storvick introduced the analytic operator-valued function space integral. Johnson and Lapidus proved that this integral can be expressed in terms of an integral of operator-valued functions [6]. In this paper, we find some operator-valued Bochner integrable functions and prove that the analytic operator-valued function space integral of a certain function is represented as the Bochner integral of operator-valued functions on some conditions.