Functional Analysis
نویسنده
چکیده
A vector space over a field K (R or C) is a set X with operations vector addition and scalar multiplication satisfy properties in section 3.1. [1] An inner product space is a vector space X with inner product 〈·, ·〉 : X ×X → K satisfying • 〈x + y, z〉 = 〈x, z〉+〈y, z〉, • 〈αx, y〉 =α〈x, y〉, • 〈x, y〉 = 〈y, x〉, • 〈x, x〉 ≥ 0 with 〈x, x〉 = 0 ⇐⇒ x = 0. [2] An inner product induces a norm on X via ‖x‖ =p〈x, x〉. [1] A normed space is a vector space X with norm ‖ ·‖ : X →R satisfying • ‖x‖ ≥ 0, • ‖x‖ = 0 ⇐⇒ x = 0, • ‖αx‖ = |α| · ‖x‖, • ‖x + y‖ ≤ ‖x‖+‖y‖. [2] Not every normed space does not comes form inner product space (e.g. l p with p 6= 2). [3] Every normed space induces a metric on X via d(x, y) = ‖x − y‖. [1] A metric space is a set X together a metric d(·, ·) : X ×X →R satisfying • 0 ≤ d(x, y) <∞,
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