1) Consider the following theorem (Theorem 4.2.12 in the lecture notes) about the comparison of locally convex topologies Theorem

Universität Konstanz

Fachbereich Mathematik und Statistik Dr. Maria Infusino

Patick Michalski

TOPOLOGICAL VECTOR SPACES–SS 2017 Exercise Sheet 9

This sheet aims to self-assess your progress and to explicitly work out more details of some of the results proposed in the lectures. You do not need to hand in solutions for these exercises but please prepare them by Friday July 7th at 10:00, when they will be discussed in F426.

1) Consider the following theorem (Theorem 4.2.12 in the lecture notes) about the comparison of locally convex topologies

Theorem. Let P = {p_i}_i∈I and Q = {q_j}_j∈J be two families of seminorms on the vector space X inducing respectively the topologies τP and τQ, which both make X into a locally convex t.v.s.. ThenτP is finer than τQ (i.e. τQ ⊆τP) iff

∀q∈ Q ∃n∈N, i1, . . . , in∈I, C >0 s.t. Cq(x)≤ max

k=1,...,npik(x),∀x∈X. (1) a) Give an alternative proof of this result without using Proposition 4.2.11 in the lecture

notes.

b) Show that the theorem still holds if we replace (1) with:

∀q∈ Q ∃n∈N, i₁, . . . , i_n∈I, C >0 s.t. Cq(x)≤

n

X

k=1

p_ik(x),∀x∈X.

2) Fix some d ∈ N. For any x = (x₁, . . . , x_d) ∈ R^d and α = (α₁, . . . , α_d) ∈ N^d₀ one defines x^α := x^α₁¹· · ·x^α_d^d. For any β ∈ N^d₀ denote by D^β the partial derivative of order |β| where

|β|:=P_d

i=1β_i, i.e.

D^β := ∂^|β|

∂x^β₁¹· · ·∂x^β_d^d = ∂^β1

∂x^β₁¹

· · · ∂^βd

∂x^β_d^d.

(a) Consider the space of infinitely differentiable functions C^∞(R^d). Show that the maps pm,K :C^∞(R^d)→Rdefined by

p_m,K(f) := sup

β∈N^d0,|β|≤m

sup

x∈K

(D^βf)(x)

form∈N0andK⊆R^dcompact are seminorms. Further, show that the topologyτP on C^∞(R^d) induced by this family of seminorms makes C^∞(R^d) into a Hausdorff l.c.t.v.s.

(b) Consider the Schwartz space or space of rapidly decreasing functions onR^d denoted by S(R^d);

S(R^d) :=

(

f ∈ C^∞(R^d) : sup

x∈R^d

x^α(D^βf)(x)

<∞,∀α, β∈N^d₀ )

.

Show that the maps qα,β :S(R^d)→Rdefined by q_α,β(f) := sup

x∈R^d

x^α(D^βf)(x)

for α, β∈N^d₀ are seminorms. Further, show that the topologyτQ onS(R^d)induced by this family of seminorms makesS(R^d) into a Hausdorff locally convex t.v.s.

(c) EndowS(R^d)⊆ C^∞(R^d)with the subspace topologyτ_P^S induced byτP. Use Exercise 1) to show that the topologyτ_Q is finer thanτ_P^S.

3) LetX be a locally convex t.v.s. whose topology is induced by a family of directed family of seminormsP. Show that a basis of neighbourhoods of the origin inX for such a topology is given by:

B_d:={r˚U_p :p∈ P, r >0}, whereU˚_p :={x∈X:p(x)<1}.

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