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##
Integral norm proving is a method used in mathematics and computer science to prove the equivalence of two mathematical expressions or functions by showing that their integrals over a certain interval are equal.
Integral norm proving is commonly used in fields such as functional analysis, real analysis, and computational mathematics to prove the equivalence of functions or expressions, especially when other methods of proof may be difficult or impossible.
Integral norm proving involves breaking down the two expressions or functions into smaller, simpler components and then showing that the integral of each component is equal. This is typically done using mathematical techniques such as substitution, integration by parts, and trigonometric identities.
Using integral norm proving can provide a more rigorous and general proof of equivalence compared to other methods. It also allows for the use of various mathematical techniques and can be applied in a wide range of fields.
One limitation of integral norm proving is that it may not always be possible to break down an expression or function into smaller components. In addition, it can be time-consuming and require a strong understanding of mathematical concepts and techniques.