Vector norm inequalities

of the Cauchy-Schwarz inequality depends only on the norm in the vector space. Notes on Vector and Matrix Norms These notes survey most important properties of norms for vectors and for linear maps from one vector space to another, and of maps norms induce between a vector space and its dual space. Dual Spaces and Transposes of Vectors Along with any space of real vectors x comes its dual space of linear functionals w T

In this note, we derive concentration inequalities for random vectors with subGaussian norm (a generalization of both subGaussian random vectors and norm bounded random vectors), which are tight up to logarithmic factors.

Proof. Suppose Ais a n nreal matrix. 1 Matrix Norms In this lecture we prove central limit theorems for functions of a random matrix with Gaussian entries. The size of a matrix is used in determining whether the solution, x, of a linear system Ax = b can be trusted, and determining the convergence rate of a vector sequence, among other things. Lemma 3.3. The most familiar norm is the Euclidean norm on Rn, which is de ned by the formula k(x 1;:::;x n)k= q x2 1 + + x2 n: De nition: Norm on a Vector Space Let V be a vector space over R. A norm on V is a function kk : V !R, denoted v 7!kvk, with the following properties: We used vector norms to measure the length of a vector, and we will develop matrix norms to measure the size of a matrix. De nition 5.1. We begin by reviewing two matrix norms, and some basic properties and inequalities.

1. In this paper, vector norm inequalities that provides upper bounds for the Lipschitz quantity ║f (T) x - f (V ) x ║ for power series f(z) = ∑ ∞ n=0 a n z n; bounded linear operators T; V on the Hilbert space H and vectors x ∈ H are established.Applications in relation to Hermite- Hadamard type inequalities and examples for elementary functions of interest are given as well. A norm is a function that measures the lengths of vectors in a vector space. Vector and Matrix Norms 5.1 Vector Norms A vector norm is a measure for the size of a vector. The last property is called the triangle inequality. The most commonly used vector norms belong to the family of p-norms, or ‘ p-norms, which are de ned by kxk p= Xn i=1 jx ijp! The triangle inequality for the `p-norm, Abstract. The operator norm of Ais de ned as kAk= sup jxj=1 kAxk; x2Rn: Alternatively, kAk= q max(ATA); where It should be noted that when n= 1, the absolute value function is a vector norm. First we need a lemma, which shows that for a complete answer it suﬃces to investigate the complex vector space C2, provided with all possible norms. A norm on a real or complex vector space V is a mapping V !R with properties (a) kvk 0 8v (b) kvk= 0 , v= 0 (c) k vk= j jkvk (d) kv+ wk kvk+ kwk (triangle inequality) De nition 5.2. Vector Norms and Matrix Norms 4.1 Normed Vector Spaces 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. We define a matrix norm in the same way we defined a vector norm.

The following two statements (1) and (2) are equivalent. 1=p: It can be shown that for any p>0, kk p de nes a vector norm.

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