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GATE Mathematics 2014 Paper

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Question : 54

Total: 55

Let X=C1[0,1]. For each f∈X , define:
p1(f):=sup{|f(t)|:t∈[0,1]}
p2(f):=sup{|f′(t)|:t∈[0,1]}
p3(f):=P1(f)+p2(f).
Which of the following statements is TRUE?

(X,p1)is a Banach space
(X,p2) is a Banach space
(X,p3) is NOT a Banach space
(X,p3) does NOT have denumerable basis

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