Topological Vector Space : Closed Kernel

Suppose that T is a topologocal vector space. Prove that a linear functional f on T is continuous if and only if ker(f) is closed.

Normed Linear Space : Continuity

Suppose that N is a normed linear space. Prove that every linear functional on N is continuous if and only if N is finite dimensional.

Two-part question on a finite dimensional normed linear space

See attachment for question. 1 Suppose that  is a finite dimensional normed linear space. a) Let be a basis for . Define Prove that 1, the closed unit ball in , is compact in (, ) b) Prove that any two norms on  are equivalent.

Banach Space and Hamel Dimension

Suppose that  is a Banach space. Prove that the Hamel dimension of  is not Please see the attached file for the fully formatted problem.

Real Analysis : Limits of Bounded Sequences

Prove if... and... are bounded sequences of real numbers, then lim sup... (See attachment for full question)

Real Analysis...Open Sets

Prove that the open interval...with a, b being real numbers is an open set. (See attachment for full question)

Real Analysis : Convergent Sequences and Limits

Prove the sequence defined by a1 = 0, a2=1 and a(n+2)= (a(n+1)+a(n))/2 for n>=0, converges and find the limit. (See attachment for full question)

One Dimensional Normed Linear Space : Completeness and Continuity

Suppose that E is a one-dimensional normed linear space. Prove that E is complete and that each linear functional on E is continous.

Functional Analysis : Continuity, Graphing

Problem: Let f be that function defined by setting (Please see the attached file for the fully formatted problem.) a. Describe graphically f(x). b. At what points is f continuous?

Analysis - Limit of the Average of the first N terms of a Sequence

Proving in Real Analysis Course. Please see the attached.