Due date: November 3
Let be a sequence of quadratic polynomials such that
Then the coefficient sequences all converge to zero.
- For , let be the space of polynomials of degree at most r. If converge uniformly to f in [0,1], then .
- The polynomials converge uniformly on [0,1], but their limit is not a polynomial function.
Let be the subspace of functions that satisfy
Then is an infinite dimensional closed subspace of .
Let be the operator given by
- is continuous with respect to the uniform norm.
- If , , and , then .
Use the previous problem to prove the following theorem: Let such that
- converges for some ;
Then converges uniformly and, if , then and .