Let {en : n = 1, 2, 3, ...} be an orthonormal basis of a complex Hilbert space H. Consider the following statements:
P: There exists a bounded linear functional f : H → ℂ such that f(en) = \(\frac{1}{n}\) for n = 1, 2, 3, ... .
Q: There exists a bounded linear functional g: H → ℂ such that g(en) = \(\frac{1}{\sqrt n}\) for n = 1, 2, 3, ... .
Then
1
both P and Q are TRUE
2
P is TRUE and Q is FALSE
3
P is FALSE and Q is TRUE
4
both P and Q are FALSE