Functional Analysis Answers

show that absolute value of. a linear functional has properties of sublinear functional

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

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. If A is a subspace of l^∞ consisting of all sequence of 0 and 1. What is the induced metric on A?

Is the real line a metric space?

Q. Which of the following sets are countable or uncountable.
(i) set of real numbers
(ii) set of complex numbers

Q. Which of the following sets are countable or uncountable.
(i)set of negative integers
(ii) set of rational numbers

Q. Which of the following sets are countable or uncountable.
(i) set of natural numbers
(ii) set of whole numbers
(iii) set of positive integers