21.9. EXERCISES 571

and

ρα (f)≡ sup{ |f (x)−f (y)||x−y|α

: x,y ∈ X , x ̸= y}.

Show that (Cα (X ;Rn) ,∥·∥α) is a complete normed linear space. This is called a

Holder space. What would this space consist of if α > 1?

10. ↑Let X be the Holder functions which are periodic of period 2π . Define Ln f (x) =Sn f (x) where Ln : X →Y for Y given in Problem 8. Show ∥Ln∥ is bounded indepen-dent of n. Conclude that Ln f → f in Y for all f ∈ X . In other words, for the Holdercontinuous and 2π periodic functions, the Fourier series converges to the functionuniformly. Hint: Ln f (x) is given by

Ln f (x) =∫

π

−π

Dn (y) f (x− y)dy

where f (x− y) = f (x)+g(x,y) where |g(x,y)| ≤C |y|α . Use the fact the Dirichletkernel integrates to one to write

∣∣∣∣∫ π

−π

Dn (y) f (x− y)dy∣∣∣∣≤

=| f (x)|︷ ︸︸ ︷∣∣∣∣∫ π

−π

Dn (y) f (x)dy∣∣∣∣

+C∣∣∣∣∫ π

−π

sin((

n+12

)y)(g(x,y)/sin(y/2))dy

∣∣∣∣Show the functions, y→ g(x,y)/sin(y/2) are bounded in L1 independent of x andget a uniform bound on ∥Ln∥. Now use a similar argument to show {Ln f} is equicon-tinuous in addition to being uniformly bounded. In doing this you might proceed asfollows. Show∣∣Ln f (x)−Ln f

(x′)∣∣≤ ∣∣∣∣∫ π

−π

Dn (y)(

f (x− y)− f(x′− y

))dy∣∣∣∣

≤ ∥ f∥α

∣∣x− x′∣∣α

+

∣∣∣∣∣∫

π

−π

sin((

n+12

)y)(

f (x− y)− f (x)− ( f (x′− y)− f (x′))sin( y

2

) )dy

∣∣∣∣∣Then split this last integral into two cases, one for |y|< η and one where |y| ≥ η . IfLn f fails to converge to f uniformly, then there exists ε > 0 and a subsequence, nksuch that

∥∥Lnk f − f∥∥

∞≥ ε where this is the norm in Y or equivalently the sup norm

on [−π,π]. By the Arzela Ascoli theorem, there is a further subsequence, Lnklf

which converges uniformly on [−π,π]. But by Problem 7 Ln f (x)→ f (x).

11. Let X be a normed linear space and let M be a convex open set containing 0. Define

ρ(x) = inf{t > 0 :xt∈M}.

Show ρ is a gauge function defined on X . This particular example is called aMinkowski functional. It is of fundamental importance in the study of locally con-vex topological vector spaces. A set, M, is convex if λx+(1−λ )y ∈M wheneverλ ∈ [0,1] and x,y ∈M.