one fourth Baseline e 21 plus integral Subscript one fourth Superscript one half Baseline e 22 plus integral Subscript one half Superscript three fourths Baseline e 23 plus integral Subscript three fourths Superscript 1 Baseline e 24 equals one fourth left-parenthesis e 21 plus e 22 plus e 23 plus e 24 right-parenthesis"/>
and, hence,
By induction, one can show that
for every
which goes to zero for sufficiently large
for every
The next result follows from Theorem 1.80. A proof of it can be found in [75, Theorem 5].
Theorem 1.85: Suppose is nonconstant on any nondegenerate subinterval of . Then, the mapping
is an isometry, that is onto a dense subspace of .
The next result, known as straddle Lemma, will be useful to prove that the space
Lemma 1.86 (Straddle Lemma): Suppose are functions such that is differentiable, with , for all . Then, given , there exists such that
whenever .
The next result is adapted from [75, Theorem 8].
Proposition 1.87: Suppose is differentiable and nonconstant on any nondegenerate subinterval of . Then, the Banach space is dense in under the Alexiewicz norm .
Proof. Assume that
