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GATE Mathematics 2013 Paper

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Question : 48

Total: 65

Let c00 be the vector space of all complex sequences having finitely many non-zero terms. Equip c00 with the inner product ❬x,y❭=

xnyn for all x=(xn) andy=(yn)‌in‌c00. Define f:c00→C by f(x)=

. Let N be the kernel of f.
Which of the following is FALSE ?

f is a continuous linear functional
‖f‖≤
π
√6
There does not exist any y∈c00 such that f(x)=❬x,y❭ for all x∈c00
N⊥≠{0}

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