Functional Analysis Basics c © 2007
نویسنده
چکیده
Normed Space [1, 2, §2]. A norm ‖·‖ on a linear space (U ,F) is a mapping ‖·‖ : U → [0,∞) that satisfies, for all u,v ∈ U , α ∈ F , 1. ‖u‖ = 0 ⇐⇒ u = 0. 2. ‖αu‖ = |α| ‖u‖. 3. Triangle inequality: ‖u+ v‖ ≤ ‖u‖+ ‖v‖. A norm defines a metric d(u,v) := ‖u− v‖ on U . A normed (linear) space (U , ‖·‖) is a linear space U with a norm ‖·‖ defined on it. • The norm is a continuous mapping of U into R+. • A norm ‖·‖ on a linear space U is said to be equivalent to a norm ‖·‖0 on U if there are positive numbers a and b such that a‖u‖0 ≤ ‖u‖ ≤ b‖u‖0 for all u ∈ U . Equivalent norms define the same topology on U . • The metric d induced by a norm is translation invariant, i.e., it satisfies ◦ d(u+ x,v + x) = d(u,v), ◦ d(αu+ αv) = |α| d(u,v) for all u,v,x ∈ U and α ∈ F . • Riesz’s Lemma: Let Y and Z be linear subspaces of a normed space U and let Y be a closed proper subset of U . Then, for every θ ∈ (0, 1), there is a z ∈ Z such that ‖z − y‖ ≥ θ for ‖z‖ = 1 and for all y ∈ Y. • A subset T of a normed space U is said to be total in U if span T is dense in U . • Let S be a linear subspace of a normed space U . If S is open as a subset in U , then S = U .
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