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normed vector space∗ rspuzio† 2013-03-21 13:17:29 Let F be a field which is either R or C. A normed vector space over F is a pair (V, k·k) where V is a vector space over F and k·k : V → R is a function such that 1. kvk ≥ 0 for all v ∈ V and kvk = 0 if and only if v = 0 in V (positive definiteness) 2. kλvk = |λ|kvk for all v ∈ V and all λ ∈ F 3. kv + wk ≤ kvk + kwk for all v, w ∈ V (the triangle inequality) The function k·k is called a norm on V . Some properties of norms: 1. If W is a subspace of V then W can be made into a normed space by simply restricting the norm on V to W . This is called the induced norm on W . 2. Any normed vector space (V, k·k) is a metric space under the metric d : V × V → R given by d(u, v) = ku − vk. This is called the metric induced by the norm k·k. 3. It follows that any normed space is a locally convex topological vector space, in the topology induced by the metric defined above. 4. In this metric, the norm defines a continuous map from V to R - this is an easy consequence of the triangle inequality. 5. If (V, h, i) is an inner p product space, then there is a natural induced norm given by kvk = hv, vi for all v ∈ V . 6. The norm is a convex function of its argument. ∗ hNormedVectorSpacei created: h2013-03-21i by: hrspuzioi version: h31604i Privacy setting: h1i hDefinitioni h46B99i † This text is available under the Creative Commons Attribution/Share-Alike License 3.0. You can reuse this document or portions thereof only if you do so under terms that are compatible with the CC-BY-SA license. 1