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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
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