By S. Zaidman

ISBN-10: 0273086618

ISBN-13: 9780273086611

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**Additional resources for Almost-periodic functions in abstract spaces**

**Example text**

Evaluate W(f) by rewriting f(t) = exp ](t) for some real-valued function] [exp s denotes e 2ltiS ]. With this rewriting, one can convert the line integral into an ordinary integral via the substitution z = f(t) = exp](t). Thus dz = z271:il'(t) dt and W(f) = -1. 2m f J -dz = z fl f'(t) - dt = f(l) - - f(O). 0 For example, let f(t) = e2ltimt be the function wrapping I around SI Iml times (counterclockwise if m ~ 0 and clockwise if m < 0). Here we may let ](t) = mt, and so W(f) = ](1) - ](0) = m. (Note that there are other possible choices for j, namely, ](t) = mt + k for any fixed integer k.

If X is a subset of R", then the affine set spanned by X in R" is the same as the affine set spanned by X in Rk. 3. ) + 2 points. (Hint: Definition. Let {Po, Pl' ... , Pm} be an affine independent subset of Rn, and let A be the affine set spanned by this subset. 4 gives a unique (m + I)-tuple (to, t l , ... , t m) with L ti = 1 and x = L:"=o tiPi. The entries of this (m + I)-tuple are called the barycentric coordinates of x (relative to the ordered set {Po, Pl' ... 2, the barycentric coordinates of a point relative to {Po, Pl' ...

For n ~ 1, show that S" is a deformation retract of the "punctured disk" D"+1 - {O}. 1. 31. Let a = (0, ... , 0, 1) and b = (0, ... , 0, -1) be the north and south poles, respectively, of S·. Show that the equator S·-l is a deformation retract of S' - {a, b}, hence S·-l and S' - {a, b} have the same homotopy type. 32. Assume that X, Y, and Z are spaces with X c Y. If X is a retract, then every continuous map f: X --+ Z can be extended to a continuous map l: y --+ Z, namely, = fr, where r: Y --+ X is a retraction.

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