Recent content by CarmineCortez

Homework Statement prove that if f is continuously differentiable on a closed interval E, then f is Lipschitz continuous on E. The Attempt at a Solution so I'm letting E be [a,b] I'm using the mean value theorem to show secant from a->b = some value, then I'm saying if I subtract...

Homework Statement Show f(x) = x^(1/3) is not lipschitz continuous on (-1,1). Homework Equations I have abs(f(x)-f(y)) <= k*abs(x-y) when I try to show that there is no K to satisfy I have problems

There is a thm that says if spectral norm <1 then A^n -> 0 as n-> infinity. and I proved above that spectral norm is <1 so I'm lost again...

(1/||x|| ) Tx = T(x/||x||)

I don't know that T is bounded...T is on R^n Tx >= T(x/||x||)

Homework Statement ||T|| = {max|T(x)| : |x|<=1} show this is equivalent to ||T|| = {max|T(x)| : |x| = 1} The Attempt at a Solution {max |T(x)| : x<=1} = {max ||x|| ||T(x/||x||)|| : |x|<=1} <= {max ||T(x)|| : |x| = 1} does that look right? I need to show equality...

I need to show: sum from k=0 to infinity of (I-T)^k converges absolutely to T^(-1) so if ||T-I|| <1 then is ||I-T|| < 1? and all the properties I listed carry over? I'm still not too sure where to go with this. when the spectral radius is <1, the higher powers of the matrix tend to 0, so it...

0 = λ*v + I*v => -1 = λ but I know my spectral radius is <1 so contradiction...

if 0 was an eigenvalue of T then T would be singular..

Homework Statement If T is a linear transformation on R^n with || T-I || < 1, prove that T is invertible. The Attempt at a Solution So a linear transformation T is invertible iff the matrix T is not singular. and I know for any matrix A, ||A|| > spectral radius(A). so, spectral...

Homework Statement I'm supposed to convert the quadratic equation into complex polar form to find the roots of a quadratic with complex constants. so b2-4ac = p*cis(phi) and (b^2-4ac)1/2 has two roots 1.p1/2cis(1/2 * phi+2pi) and 2. p1/2(phi/2) so I've subbed everything into the equation...

Homework Statement I have to show that l1, l2 and linfinity are norms The Attempt at a Solution Do you just go through the conditions for norm spaces ie: 1. ||x||>0, ||x|| = 0 iff x = 0 2.triangle inequality 3.||cx|| < |c|||x|| if the space satisfies these conditions it is a norm??

Homework Statement suppose f and g are uniformly continuous functions on X and f and g are bounded on X, show f*g is uniformly continuous. The Attempt at a Solution I know that if they are not bounded then they may not be uniformly continuous. ie x^2 and also if only one is bounded...

Homework Statement if f and g are 2 uniformly continuous functions on X --> R show that f+g is uniformly continuous on X The Attempt at a Solution I tried showing that f+g is Lipschitz because all Lipschitz functions are uniformly continuous. So i end up with d(x_1,x_2) <...

Homework Statement Let f map [a,b]-->R be a continuous nonegative function. Suppose Integral f(x)dx from a to b = 0 show that f = 0 on [a,b] The Attempt at a Solution Just not sure if this is good or not.. so the lower sum <= 0 = integral f(x) dx but the lower sum must be 0...