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In mathematics, a norm is a function from a real or complex vector space to the non-negative real numbers that behaves in certain ways like the distance from the origin: it commutes with scaling, obeys a form of the triangle inequality, and is zero only at the origin.
- 1.1 Infinite-dimensional vector spaces
- L∞([0, 1]) ⊂ L2([0, 1]) ⊂ L1([0, 1])
- 1.3 Linear maps
Vector spaces are defined by the usual axioms of addition and scalar multiplication. The important spaces are as follows. Note that there are real-valued versions of all of these spaces.
Note that this nesting doesn’t hold for Lp(R). There is no constant K such that for all
Suppose U and V are normed spaces; Consider the set of all possible linear maps
1 C 2 kxk a kxk a0 C 2 C 1 kxk a; andhencekk a andkk a0 areequivalent. Q.E.D.1 1While transitivity establishes upper/lower bounds for the relationship between kk aand 0, and hence their equivalence, the constants C 0 1 C2 and C 0 2 C1 are not in general the tightest possible bounds even if the constants C 1;2 and C0 1;2 relating them to kk 1 ...
- 264KB
- 2
The role of a norm. norm on a real or complex vector space provides a special kind of metric, one which is compatible with the linear structure. In particular addition and scalar multiplication are continuous maps; see 0.8. norm provides a measure of distance.
Jun 24, 2022 · The 1-Norm and the 2-Norm are P-Norms, where P=1 and P=2, respectively. We choose the values of one and two because they are commonly used throughout applications, but P can be set to any number greater than one.
- Travis Cooper
By a normed linear space (briefly normed space) is meant a real or complex vector space E in which every vector x is associated with a real number | x |, called its absolute value or norm, in such a manner that the properties (a′) − (c′) of §9 hold. That is, for any vectors x, y ∈ E and scalar a, we have. (i) | x | ≥ 0;
p-norm. Proposition 4.1. If E is a finite-dimensional vector space over R or C, for every real number p ≥ 1, the p-norm is indeed a norm. The proof uses the following facts: If q ≥ 1isgivenby 1 p + 1 q =1, then (1) For all α,β ∈ R,ifα,β ≥ 0, then αβ ≤ αp p + βq q. (∗) (2) For any two vectors u,v ∈ E,wehave n i=1 |u ...
