The discussion revolves around proving that certain integral norms, specifically ||f||1 = ∫|f| from 0 to 1 and ||f|| = ∫t|f(t)|dt, define norms on the space of continuous functions C[0,1]. Participants are exploring the necessary conditions for these norms and how to demonstrate their validity.
Some participants have made progress on the first norm but express uncertainty regarding the second norm involving t|f(t)|. There is an ongoing exploration of inequalities related to integrals and how they apply to the norms being discussed.
Participants are considering the implications of continuity and the properties of integrals, particularly in relation to the norms defined on C[0,1]. There is a focus on ensuring that certain values remain positive within the context of the proofs.
cummings12332 said:Homework Statement
show that ||f||1 = ∫|f| (integral from 0 to 1) does define a norm on the subspace C[0,1] of continuous functions
and also the same for ||f||= ∫t|f(t)|dt is a norm on C[0,1]Homework Equations
(there are 3 conditions , i just don't know how to prove that ||v||>0,||v||=0 implies v=0)
LCKurtz said:If a continuous function is nonzero (or positive) at a point, what about its value nearby that point, and why?
cummings12332 said:yes if it is not zero ,then |f(x)-f(c)|<esillope then choose esillope to be f(c)/2 then it will get |f(x)|>sth... but how can u ensure that f(c) >0 for esillope has to be >0
LCKurtz said:You have this property of integrals to work with: If ##f,g\in C[a,b]## and ##f(x)> g(x)## on [a,b] then ##\int_a^b f(x)\, dx > \int_a^b g(x)\, dx##