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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 semi-norm is called a Semi-normed 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 '''semi-norm 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'', # ''p''(v) ≥ 0 (''positivity'') | ||
| Every | "http://wwwinformationdelightinfo/encyclopedia/entry/linear_form" class="copylinks">Linear Form ''f'' on a vector space defines a semi-norm 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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