Functional Analysis - Normed Linear Space
Suppose that N is a normed linear space. Prove that each finite dimensional linear submanifold of N is complete and therefore closed.
Linear functional on a topological vector space
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
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
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Real Analysis.....im sup Inequalities
Prove if... and... are bounded sequences of real numbers, then lim sup... (See attachment for full question)
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
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 ATTACHMENT FOR SETTING) a. Describe graphically f(x). b. At what points is f continuous? (This is a graduate course in real analysis. It seems that so many have attempted to solve this question but to no avail! Please give it a try).
