| Normed Vector Space |
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# The zero vector, 0, has zero length; every other vector has a positive length. # Multiplying a vector by a positive number changes its length without changing its direction. See Unit Vector . # The Triangle Inequality holds. That is, taking norms as distances, the distance from A through B to C is never shorter than going directly from A to C, or the shortest distance between any two points is a straight line. Their generalization for more abstract Vector Space s, leads to the notion of Norm . A vector space on which a norm is defined is then called a '''normed vector space'''. DEFINITION A semi normed vector space is a Pair (''V'',''p'') where ''V'' is a Vector Space and ''p'' a Semi Norm on ''V''. | ||
| If (''V'', ·) Is A Normed Vector Space, The Norm · Induces A Notion Of ''distance'' And Therefore A | "http://wwwinformationdelightinfo/information/entry/topology" class="copylinks">Topology on ''V'' This distance is defined in the natural way: the distance between two vectors '''u''' and '''v''' is given by '''u'''-'''v''' This topology is precisely the weakest topology that makes · continuous Furthermore, this natural topology is compatible with the linear structure of ''V'' in the following sense: |
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| Similarly, For Any Semi-normed Vector Space We Can Define The Distance Between Two Vectors '''u''' And '''v''' As '''u'''-'''v''' This Turns The Semi Normed Space Into A | "http://wwwinformationdelightinfo/information/entry/semi_metric_space" class="copylinks">Semi Metric Space (notice this is weaker than a metric) and allows the definition of notions such as Continuity and Convergence |
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| An ''isometry'' Between Two Normed Vector Spaces Is A Linear Map ''f'' Which Preserves The Norm (meaning ''f''('''v''') | '''v''' for all vectors '''v''') Isometries are always continuous and Injective A Surjective isometry between the normed vector spaces ''V'' and ''W'' is called a ''isometric isomorphism'', and ''V'' and ''W'' are called ''isometrically isomorphic'' Isometrically isomorphic normed vector spaces are identical for all practical purposes |
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| When Speaking Of Normed Vector Spaces, We Augment The Notion Of | "http://wwwinformationdelightinfo/information/entry/dual_space" class="copylinks">Dual Space to take the norm into account The dual ''V'' ' of a normed vector space ''V'' is the space of all ''continuous'' linear maps from ''V'' to the base field (the complexes or the reals) &mdash such linear maps are called "functionals" The norm of a functional &phi is defined as the Supremum of &phi('''v''') where '''v''' ranges over all unit vectors (ie vectors of norm 1) in ''V'' This turns ''V'' ' into a normed vector space An important theorem about continuous linear functionals on normed vector spaces is the Hahn-Banach Theorem |
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