34 2. The First Variation The supremum (or least upper bound) is here in case we are work- ing with functions that are piecewise continuously differentiable. If our functions are continuously differentiable, the supremum can be replaced by a maximum. We will use the weak and strong norms to establish neighborhoods in function space. Weak and strong norms permit different variations about the optimal solution. Since the weak norm does not impose any restriction upon the derivative, an -neighborhood in a weakly- normed space will include strong variations (see Figure 2.3) that differ significantly from the optimal solution in slope while remaining close in ordinate. Strong variations may have arbitrarily large derivatives. Example 2.2. The function h(x) = sin x 2 (2.23) never exceeds and yet its derivative, h (x) = 1 cos x 2 , (2.24) may become arbitrarily large as is made small. x y a b y (x ) = ˆ(x ) + h(x ) Figure 2.4. Weak variations

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