For 0<α<1, C^α[0,1] is the Hölder Banach space with norm |f(0)|+sup_(s≠t)|f(t)−f(s)|/|t−s|^α. T:C^α→C[0,1] is inclusion into sup norm. Which statements hold?
Correct answer: (A) T is compact linear.; (C) Its image is dense.
Explanation
For norm-bounded inputs, |f(t)|≤|f(0)|+[f]_α≤‖f‖_α and |f(t)−f(s)|≤C|t−s|^α. Arzelà–Ascoli gives relative compactness, proving A and boundedness. Polynomials are Hölder and uniformly dense, proving C. The image is proper: t^β with 0<β<α is continuous but not α-Hölder, so the dense image is not closed. **A,C**.
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