| E. T. Copson - Mathematics - 1988 - 152 pages
...defined, for each point x of X, a real number called the norm of x and denoted by ||:r||, such that **(i) ||x|| ^ 0 and ||x|| = 0 if and only if x = 0. (ii)** For each scalar multiplier a and for each point x of X, (iii) H For example, in the space just referred... | |
| Peng Yee Lee - Mathematics - 1989 - 179 pages
...functions on [a,b]. A normed linear space E is a linear space provided with a norm ||x|| satisfying **the following properties: (i) ||x|| > 0 and ||x|| - 0 if and only if x - 0; (ii)** ||x + y|| < ||x|| + (y||; (iii) 1 ax|| - | a | ||x|| for real a. A normed linear space E is complete... | |
| Krzysztof Kowalski, W.-H. Steeb - Science - 1991 - 184 pages
...Let B be a vector space over f, where f = K or C. Definition: A norm on B is a function (D such that **(i) ||x|| > 0 and ||x|| = 0 if and only if x = 0 (ii)** ||z + y|| < ||x|| + \\y\\ for x,y € B (Hi) \\Xx\\ = |A|||z|| for A € F, x e B. Definition: A sequence... | |
| Krzysztof Kowalski, W.-H. Steeb - Science - 1991 - 184 pages
...Let B be a vector space over f, where f = H or C. Definition: A norm on B is a function IH|:8->ft (1) **(i) ||x|| > 0 and ||x|| = 0 if and only if x = 0** («) ||x + y|| < ||x|| + W for x,ytB (iii) ||Ax|| = |A|||x|| for A 6 F, x 6 B. Definition: A sequence... | |
| Myron B. Allen, Eli L. Isaacson - Mathematics - 1998 - 492 pages
...mapping || • ||: K" -» E to be a norm, it must satisfy three conditions: (i) For any vector xe R", **||x|| > 0, and ||x|| = 0 if and only if x = 0. (ii)** Whenever x 6 M" and c 6 K, ||cx|| = |c|||x||. (iit) Whenever x, ye K", ||x + y||< ||x|| + ||y||. An... | |
| Qazi Ibadur Rahman, Gerhard Schmeisser - Mathematics - 2002 - 742 pages
...• || : C" -> K with the following properties: For x, y € C" and ae C, we have (i) ||x|| > 0 when **x ^ 0, and ||x|| =0 if and only if x = 0 : (ii)** l|ax|| = M ' 1MI ; Three familiar vector norms are: (1) the I1 norm \\x\\i := £"=1 \xv\ ; (2) the... | |
| Don Hong, Jianzhong Wang, Robert Gardner - Mathematics - 2004 - 392 pages
...is also called the norm of x in general and denoted by||x||. Theorem 1.4.1 The length |x| satisfies: **(i) |x| > 0 and |x| = 0 if and only if x = 0** = (0, • • • , 0). (ii) For all A € R, |Ax| = |A| |x|. (iii) For all x, ye R", we have |x +... | |
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