Advanced Stuff (26 problems)

Letbe a sequence of linearly independent vectors in an inner product space. Define the vectors inductively as

Show thatis an orthonormal sequence having the same closed linear span as the x_j’s.

Find a vector inorthogonal to (1, 1, 1) and, where.

Let E be a Banach space and let. Define

by. Show that T is a linear operator and is bounded with respect to the operator norm on, with.

Solution 7.2: We have that, and sinceandare linear, thenis linear. On the other hand

The last inequality implies that. But also,

which implies that. This proves the result.

Prove that the closure of a convex set in a normed space is convex.

The Volterra operatoronis the operator of the indefinite integration

, 0<t<1

Show that V is a bounded operator.

Let be a complete orthonormal sequence in a Hilbert space H and letbe a complex sequence. Show that there is a bounded linear operator D on H such that for all n if and only if is a bounded sequence. What is when defined?

Prove that the open and closed unit balls in a normed space are convex

The first 3 Legendre Polynomials are

Show that if a > 0, then the convergence of the sequence in the above problem (exercise 2) is uniform on the interval [a,∞), but is not uniform on the interval [0,∞).

Show that the sequence

doesn’t converge uniformly on [0, 2].

Show that ifandconverge uniformly on the set A toandrespectively, thenconverges uniformly to.

Suppose the sequence converges uniformly to f on the set A and suppose that each f_n is bounded on A. Show that the function is bounded on A.

Suppose thatis differentiable at c and that. Show thatis differentiable at c if and only if

Show that the functionis uniformly continuous on the real line.