316 CHAPTER 12. BANACH SPACES

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 6 Ln f (x)→ f (x).

10. Let X be a normed linear space and let M be a convex open set containing 0. Defineρ(x) = inf{t > 0 : x

t ∈M}. Show ρ is a gauge function defined on X . This particularexample is called a Minkowski functional. It is of fundamental importance in thestudy of locally convex topological vector spaces. A set M, is convex if λx+(1−λ )y ∈M whenever λ ∈ [0,1] and x,y ∈M.

11. ↑The Hahn Banach theorem can be used to establish separation theorems. Let M bean open convex set containing 0. Let x /∈ M. Show there exists x∗ ∈ X ′ such thatRex∗(x) ≥ 1 > Rex∗(y) for all y ∈ M. Hint: If y ∈ M,ρ(y) < 1. Show this. Ifx /∈M, ρ(x)≥ 1. Try f (αx) = αρ(x) for α ∈ R. Then extend f to the whole spaceusing the Hahn Banach theorem and call the result F , show F is continuous, then fixit so F is the real part of x∗ ∈ X ′.

12. A Banach space is said to be strictly convex if whenever ∥x∥ = ∥y∥ and x ̸= y, then∥∥ x+y2

∥∥ < ∥x∥. F : X → X ′ is said to be a duality map if it satisfies the following:a.) ∥F(x)∥ = ∥x∥. b.) F(x)(x) = ∥x∥2. Show that if X ′ is strictly convex, thensuch a duality map exists. The duality map is an attempt to duplicate some of thefeatures of the Riesz map in Hilbert space. This Riesz map R is the map whichtakes a Hilbert space to its dual defined as follows: R(x)(y) = (y,x) . The Rieszrepresentation theorem for Hilbert space says this map is onto. Hint: For an arbitraryBanach space, let

F (x)≡{

x∗ : ∥x∗∥ ≤ ∥x∥ and x∗ (x) = ∥x∥2}