X={f∈C[0,1]:f(0)=f(1)=0} with sup norm; Y=C[0,1] with L² norm. On their closed unit balls U_X,U_Y, define Tf=Sf=∫₀¹f(t)dt. S1: sup over U_X of |Tf| is attained. S2: sup over U_Y of |Sf| is attained. Choose the truth values.
In X the supremum is 1: continuous plateau functions with narrow endpoint ramps approach it, but any function bounded by 1 and zero at both endpoints has strictly smaller absolute integral. S1 is false. In Y, Cauchy–Schwarz gives |∫f|≤‖f‖₂≤1, attained by f≡1. S2 is true. **B**.
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