Let T = [0, 2). a) Prove that the function (x) := (cos x, sin x) is

Let T = [0, 2Ï€).
a) Prove that the function
Φ(x) := (cos x, sin x)
is 1-1 from T onto Ï‘B1(0, 0) Š‚ R2.
b) Prove that
p(x, y) := ||Φ (x) - Φ(y)||
is a metric on T.
c) Prove that a function f is continuous on (T, p) if and only if it is continuous and periodic on [0, 2Ï€]; that is, if and only if f has an extension to [0, 2Ï€) which is continuous in the usual sense which also satisfies f(0) = f(2Ï€).
d) A function P is called a trigonometric polynomial if

for some scalars ak, bk. Prove that given f ˆˆ C(T) there is a sequence of trigonometric polynomials Pn such that Pn †’ f uniformly on [0, 2Ï€] as n †’ ˆž.

Transcribed Image Text:

P(x)-Σ ak cos(kx) + bk sin(kx) に0