(a) Prove that the space R consisting of all infinite sequences x = (x1,x2,x3,...) of real numbers...

Question:

(a) Prove that the space Rˆž consisting of all infinite sequences x = (x1,x2,x3,...) of real numbers Xi ˆˆ R forms a vector space.
(b) Prove that the set of all sequences x such that

Form a subspace. Commonly de- noted „“2 Š‚ Rˆž.
(c) Write down two examples of sequences x belonging to „“2 and two that do not belong to „“2.
(d) True or false: If x ˆˆ „“2, then xk †’ 0 and k †’ˆž.
(e) True or false: If xk †’ 0 as k †’ ˆž then x ˆž „“2.
(f) Let a be fixed, and let x be the sequence with xk = ak. For which values of Î± is x ˆˆ „“2?
(g) Answer part (f) when xk = ka.
(h) Prove that

defines an inner product on the vector space t1. What is the corresponding norm?
(i) Write out the Cauchy-Schwarz and triangle inequalities for the inner product space l2.