Abstract
Let X be a vector space over a basic field 𝔽 and let p: X → ℝ be a seminorm. Then
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(1)
dom p is a subspace of X;
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(2)
p(x) ≥ 0 for all x ∈ X;
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(3)
the kernel ker p≔ {p = 0} is a subspace in X;
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(4)
the sets \({{\mathop{B}\limits^{ \circ } }_{p}}: = \left\{ {p < 1} \right\}\) and B p ≔ {p ≤ 1} are absolutely convex; moreover, p is the Minkowski functional of every set B such that \({{\mathop{B}\limits^{ \circ } }_{p}} \subset B \subset {{B}_{p}};\)
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(5)
X = dom p if and only if \({{\mathop{B}\limits^{ \circ } }_{p}}\) is absorbing.
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© 1996 Springer Science+Business Media Dordrecht
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Kutateladze, S.S. (1996). Multinormed and Banach Spaces. In: Fundamentals of Functional Analysis. Kluwer Texts in the Mathematical Sciences, vol 12. Springer, Dordrecht. https://doi.org/10.1007/978-94-015-8755-6_5
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DOI: https://doi.org/10.1007/978-94-015-8755-6_5
Publisher Name: Springer, Dordrecht
Print ISBN: 978-90-481-4661-1
Online ISBN: 978-94-015-8755-6
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