Functional Analysis
نویسنده
چکیده
Definition 1.2 A real vector space X is called a real normed space if there exists a map || · || : X → IR, such that, for all λ ∈ IR and x, y ∈ X, (i) ||x|| = 0⇔ x = 0; (ii) ||λx|| = |λ| ||x||; (iii) ||x + y|| ≤ ||x|| + ||y|| (triangle inequality). The map || · || is called the norm. If || · || only satisfies (ii) and (iii) then it is called a seminorm. Note that X is automatically a metric space with respect to the metric defined by d(x, y) = ||x− y||. Exercise 1 In IR we consider for x = (x1, x2) the norms ||x||1 = |x1|+ |x2|, ||x||2 = √ x1 + x 2 2, ||x||∞ = max{|x1|, |x2|}.
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