For an orthonormal basis (e_n) of a complex Hilbert space H: P:there is a bounded linear f:H→ℂ with f(e_n)=1/n. Q:there is a bounded linear g:H→ℂ with g(e_n)=1/√n. Choose truth values.
By the Riesz representation theorem, the basis coefficients of a bounded functional must be square summable, and any square-summable coefficients define one. Σ1/n² converges but Σ1/n diverges. Thus f exists and g does not. **B**.
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