| Norm (mathematics) |
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Information AboutNorm (mathematics) |
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A simple example is the 2-dimensional Euclidean Space R2 equipped with the Euclidean Norm . Elements in this vector space (e.g., (3, 7) ) are usually drawn as arrows in a 2-dimensional Cartesian Coordinate System starting at the origin (0, 0). The Euclidean norm assigns to each vector the length of its arrow. A vector space with a norm is called a Normed Vector Space . Similarly, a vector space with a seminorm is called a Seminormed Vector Space . DEFINITION Given a Vector Space ''V'' over a Subfield F of the Complex Number s such as the complex numbers themselves or the Real or Rational Number s, a '''seminorm on''' ''V'' is a Function ''p'':''V''→'''R'''; ''x''→ ''p''(''x'') with the following properties: For all ''a'' in ''F'' and all u and '''v''' in ''V'', | ||
| Although Every Vector Space Is Seminormed (eg, With The Trivial Seminorm In The Examples Section Below), It May Not Be Normed Every Vector Space ''V'' With Seminorm ''p''('''v''') Induces A Normed Space ''V/W'', Called The | "http://wwwinformationdelightinfo/information/entry/quotient_space" class="copylinks">Quotient Space , where ''W'' is the subspace of ''V'' consisting of all vectors '''v''' in ''V'' with ''p''('''v''') = 0 The induced norm on ''V/W'' is given by ''W''+'''v''' = ''p''('''v''') and is clearly well-defined |
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| Every | "http://wwwinformationdelightinfo/information/entry/linear_form" class="copylinks">Linear Form ''f'' on a vector space defines a seminorm by '''x''' → ''f''('''x''') |
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| :<math>\\mathbf{x}\ : | \sqrt{x_1^2 + \cdots + x_n^2}</math> |
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| :<math>\\mathbf{z}\ : | \sqrt{z_1^2 + \cdots + z_n^2}</math>, equivalent with the Euclidean norm on '''R'''<sup>2''n''</sup> |
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| :<math>\x\ : | 2x_1 + \sqrt{3x_2^2 + \max(x_3,2x_4)^2}</math> |
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| :<math> \x\ P | \left(\sum_{i\in\mathbb N}x_i^p
ight)^{ rac1p} </math> resp <math> \f\_{p,X} = \left(\int_Xf(x)^p\,\mathrm dx
ight)^{ rac1p} </math> |
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| Any | "http://wwwinformationdelightinfo/information/entry/inner_product" class="copylinks">Inner Product induces in a natural way the norm <math>\x\ := \sqrt{\langle x,x
angle}</math> |
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| :<math>x^ Op Y\le\ X\ P\y\ Q\qquad Rac{1}{p}+ Rac{1}{q} | 1</math>1 |
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