When X is a compact Hausdorff space and E is a real Banach space there is a considerable literature on extremal properties of the space C ( X , E ) of continuous E -valued functions on X . What happens if the Banach spaces in which the functions on X take their values vary over X ? In this paper, we obtain some extremal results on the section space Γ ( π ) and its dual Γ ( π ) * of a real Banach bundle π : ℰ → X (with possibly varying fibers), and point out the difficulties in arriving at totally satisfactory results.