Problem 1. Let || · || be a norm in R". Prove that the function f(x) = ||x|| is convex.Hints:• Implicit in the definition is that the norm is defined for all vectors. So, you only need to check Jensen'sinequality.Since you are only given a generic norm (not some particular norm), you'll need to make use of theproperties in the definition of norm.• One helpful thing to keep in mind is that since 0 lies in the interval [0, 1], so does 1 – 0. That meansboth 0 and 1– 0 are non-negative. That's useful to know when the terms |0| and |1 – 0| show up. In this assignment, you'll practice proving statements about convexity for higher dimensional functions. Youcan find the definition of convexity in the handout for this module.We’ll start with the convexity of norms.Definition 1. A function ƒ : R" → R is called a norm if the following conditions are true.1. f(x) > 0 for all x E R",2. f(x) = 0 implies that x =0,3. f(x + y) < f (x)+ f(y) for all x, y E R", and4. f(ax) = |a|f(x) for all x E R" and all a E R.The usual notation in linear algebra is to use ||x|| to represent a norm of x rather than f(x). So, for example,item #3 in the list above (called the triangle inequality) can be written as ||x + y|| < |||| + ||y||. It's alsocommon to see notation like "Consider a norm || · || in R"." The dot in the notation is just a placeholder forthe argument to the norm. It basically means, "put input here."Norms are a generalization of distance. Some important norms are• the 2-norm (or Euclidean norm)= 레레• the 1-normn||r||1 = :|,i=1• the infinity norm (or max norm)||x|| = max{|x1|, |x2], ... , |xn|}-

Problem 1. Let || · || be a norm in R". Prove that the function f(x) = ||x|| is convex. Hints: • Implicit in the definition is that the norm is defined for all vectors. So, you only need to check Jensen's inequality. • Since you are only given a generic norm (not some particular norm), you’ll need to make use of the properties in the definition of norm. • One helpful thing to keep in mind is that since 0 lies in the interval (0, 1], so does 1 – 0. That means both 0 and 1 – 0 are non-negative. That's useful to know when the terms |0| and |1 – 0| show up.

Problem 1. Let || · || be a norm in R". Prove that the function f(x) = ||x|| is convex. Hints: • Implicit in the definition is that the norm is defined for all vectors. So, you only need to check Jensen's inequality. • Since you are only given a generic norm (not some particular norm), you’ll need to make use of the properties in the definition of norm. • One helpful thing to keep in mind is that since 0 lies in the interval (0, 1], so does 1 – 0. That means both 0 and 1 – 0 are non-negative. That's useful to know when the terms |0| and |1 – 0| show up.

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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