Abstract
For a real Banach space (written B space hereafter) E we let B(E) denote the unit ball:
For example B(R) is the closed interval [−1, 1]. For a bounded linear operator T: E 1 → E 2 between B spaces, the operator norm of T can be described as:
Of course by linearity \(\left| {T\left( x \right)} \right| \leqslant \left\| T \right\|\left| x \right|\) for all x ∈ E l. The set L(E 1, E 2) of all such bounded linear operators is a B space with the operator norm, and its unit ball is the set of operators of norm at most 1. Equivalently:
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© 1997 Springer Science+Business Media New York
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Akin, E. (1997). Compactifications. In: Recurrence in Topological Dynamics. The University Series in Mathematics. Springer, Boston, MA. https://doi.org/10.1007/978-1-4757-2668-8_6
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DOI: https://doi.org/10.1007/978-1-4757-2668-8_6
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