Banach Space

Let I = [a,b] be a finite interval. Show that the space C(I,R^n) of continous functions from I into R^n is a Banach space with the uniform norm llull = sup l u(t) l where t is in I. (Show that this is a norm and that C(I,R^n) is complete) See attached file. Please be very detailed when answering question.

Prove Abel's formula for the Wronskian.

Prove Abel's formula for the Wronskian. Hint: First show that the derivative of a p by p determinant is the sum of p determinants, each of which has only one row differentiated.

Poisson Formula

(See attached file for full problem description)

Problem Set

(See attached file for full problem description) a) Determine the irreducibility of x20-11 over Q(set of rationals), and use it to prove or disprove that the ideal is a maximal ideal of Q[x]. b) Construct an integral domain R and an element a in R such that a is irreducible but not prime in R. c) Suppose that R is a principal ideal domain and a in R is irreducible. If a does not divide b in R, prove that a and b are relatively prime. d) Suppose p in N(set of naturals) is a prime number. Show that every element a in Zp has a p-th root, i.e. there is b in Zp with a=bp.

Series convergence

Prove that the series Sigma (k = 0 to inf) k!/k^k converges