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A Continuous, Nowhere-Differentiable Function with a Dense Set of Proper Local Extrema
In this paper, we use the following scheme to construct a continuous, nowhere-differentiable function ๐ which is the uniform limit of a sequence of sawtooth functions ๐โ : [0, 1] โ [0, 1] with increasingly sharp teeth. Let ๐ = [0, 1] x [0, 1] and ๐น(๐) be the Hausdorff metric space determined by ๐. We define contraction maps ๐คโ , ๐คโ , ๐คโ on ๐. These maps define a contraction map ๐ค on ๐น(๐) via ๐ค(๐ด) = ๐คโ(๐ด) โ ๐คโ(๐ด) โ ๐คโ(๐ด). The iteration under ๐ค of the diagonal in ๐ defines a sequence of graphs of continuous functions ๐โ. Since ๐ค is a contraction map in the compact metric space ๐น(๐), ๐ค has a unique fixed point. Hence, these iterations converge to the fixed point-which turns out to be the graph of our continuous, nowhere-differentiable function ๐. Chapter 2 contains the background we will need to engage our task. Chapter 3 includes two results from the Baire Category Theorem. The first is the well known fact that the set of continuous, nowhere-differentiable functions on [0,1] is a residual set in ๐ถ[0,1]. The second fact is that the set of continuous functions on [0,1] which have a dense set โฆ
Date:
December 1993
Creator:
Huggins, Mark C. (Mark Christopher)
System:
The UNT Digital Library
Weak and Norm Convergence of Sequences in Banach Spaces
We study weak convergence of sequences in Banach spaces. In particular, we compare the notions of weak and norm convergence. Although these modes of convergence usually differ, we show that in โยน they coincide. We then show a theorem of Rosenthal's which states that if {๐โ} is a bounded sequence in a Banach space, then {๐โ} has a subsequence {๐'โ} satisfying one of the following two mutually exclusive alternatives; (i) {๐'โ} is weakly Cauchy, or (ii) {๐'โ} is equivalent to the unit vector basis of โยน.
Date:
December 1993
Creator:
Hymel, Arthur J. (Arthur Joseph)
System:
The UNT Digital Library