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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
We will show that it is sufficient for to prove that k ka is equivalent to k k1, because norm equivalence is transitive: if two norms are equivalent to k k1, then they are equivalent to each other. In particular, suppose both k ka and k ka0 are equivalent to k k1 for constants 0 < C1 C2 and. 0 < C0.
- 264KB
- 2
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;
In fact, the 0 -norm is also known as the norm. We are now ready to formally define local minima of a functional. Let be a vector space of functions equipped with a norm , let be a subset of , and let be a real-valued functional defined on (or just on ).
Functional Analysis I studies normed spaces in general and complete normed spaces (called Banach spaces) in particular. Such spaces|principally in nite-dimensional ones|form the backbone of a theory that underpins much of applied analysis as well as being worthy of study in its own right.
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 ...
