Functional Analysis Answers

A and B are subsets of R where f:A-->B is a function.Prove that f is a bijection if and only if f^-1:B-->A(f inverse from B to A) exists as a bijection

Q. Prove shortly that normed linear space is also a metric space.

Solve the given question shortly
Q. Show that another metric D on a set S is defined by d(x,y)=∫_a^b▒〖|x(t)-y(t)|dt〗

Q. If A is a subspace of l^∞ consisting of all sequence of 0 and 1. What is the induced metric on A? (solve it shortly)

Q. Find all metric on a set X consisting of two points. (solve it shortly)

Let D be a metric on X determined all constant k s.t
(k+d) is a metric. Give its Hint or prove it shortly.

Q. State and prove Hahn Banach theorem.

Q. State and prove Zorn, s Lemma.

Q. Prove that |d(x,z)-d(y,z)|≤d(x,y)

Q. Show that another metric D on a set S defined by
d(x,y) = ∫_a^b▒〖| x(t)-y(t) |dt 〗