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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
Example 2 The norm of a diagonal matrix is its largest entry (using absolute values): A = 2 0 0 3 has norm kAk= 3. The eigenvector x = 0 1 has Ax = 3x. The eigenvalue is 3. For this A (but not all A), the largest eigenvalue equals the norm. For a positive definite symmetric matrix the norm is kAk= λmax(A).
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
Mathematically, the existence of absolute values in \(E\) amounts to that of a map (called a norm map) \(x \rightarrow|x|\) on \(E,\) i.e., a map \(\varphi : E \rightarrow E^{1},\) with function values \(\varphi(x)\) written as \(|x|,\) satisfying the laws (i)-(iii) above. Often such a map can be chosen in many ways (not necessarily via dot ...
A norm on V is a real-valued function ||·|| on V satisfying the following axioms: Positivity: ||0||=0, and || v || is a positive real number for all nonzero vectors v . Homogeneity: || cv || = |c| || v || for all scalars c and vectors v . Subadditivity: || v + w || <= || v || + || w || for all vectors v, w in V .
