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The concept of unit circle (the set of all vectors of norm 1) is different in different norms: for the 1-norm, the unit circle is a square oriented as a diamond; for the 2-norm (Euclidean norm), it is the well-known unit circle; while for the infinity norm, it is an axis-aligned square.
- 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
The norm of A is the largest ratio kAxk kxk: kAk = max x6=0. kAxk kxk. (3) . kAxk kxk is never larger than kAk (its maximum). This says that kAxk ≤ kAk kxk. Example 1 If A is the identity matrix I, the ratios are Therefore = 1. If. A kxk kxk. is an orthogonal matrix Q, kIk lengths are again preserved: kQxk = The ratios still give kQk = 1.
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 order to define how close two vectors or two matrices are, and in order to define the convergence of sequences of vectors or matrices, we can use the notion of a norm. Recall that R. += {x ∈ R | x ≥ 0}. Also recall that if z = a + ib ∈ C is a complex number, with a,b ∈ R,thenz = a−ib and |z| = √ a2+b2. (|z| is the modulus of z). 207.
Section 1.3 Matrix Norms ¶ 1.3.1 Of linear transformations and matrices; 1.3.2 What is a matrix norm? 1.3.3 The Frobenius norm; 1.3.4 Induced matrix norms; 1.3.5 The matrix 2-norm; 1.3.6 Computing the matrix 1-norm and \(\infty\)-norm; 1.3.7 Equivalence of matrix norms; 1.3.8 Submultiplicative norms
Notes on the equivalence of norms. Steven G. Johnson, MIT Course 18.335. Created Fall 2012; updated October 28, 2020. If we are given two norms k ka and k kb on some finite-dimensional vector space V over C, a very useful fact is that they are always within a constant factor of one another.
