March 1st, 2017, 11:58 PM  #1 
Newbie Joined: Mar 2017 From: tbilisi Posts: 1 Thanks: 0  metric space
How to showed that $\displaystyle \rho(f,g)=\int_{a}^{b}f(x)g(x)^p dx$ defined metric on $\displaystyle L^p[a,b]$?

March 2nd, 2017, 07:44 AM  #2 
Senior Member Joined: Mar 2015 From: New Jersey Posts: 1,297 Thanks: 94 
Step 1: Define $\displaystyle L^{p} [a,b]$

March 2nd, 2017, 11:58 AM  #3 
Senior Member Joined: Aug 2012 Posts: 1,779 Thanks: 482  Zylo I see that you're starting to channel my own pickiness. Happy to see that In this case there is a great subtlety lurking. We typically think of $\displaystyle L^p[a,b]$ as the set of functions $\displaystyle f : [a,b] \to \mathbb R$ such that $\displaystyle \int_{[a,b]} f(x) \ \mathrm dx < \infty$. I'm notating the integral that way to indicate that this is the Lebesgue and not the Riemann integral. To show this induces a metric we need to prove (among other things) that $\displaystyle d(f, g) = 0 $ if and only if $\displaystyle f = g$. But now consider $\displaystyle f(x) = 1$ and $\displaystyle g(x) = 1$ if $x$ is irrational and $\displaystyle g(x) = 0$ if $\displaystyle x$ is rational. Clearly $\displaystyle f \neq g$ since these two functions differ on the rationals, but $\displaystyle d(f,g) = 0$. What happened? The answer is that when we put on our picky hats, $\displaystyle L^p[a,b]$ is the set of equivalence classes of functions where we say that $\displaystyle f$ and $\displaystyle g$ are equivalent if $\displaystyle f$ and $\displaystyle g$ differ on a set of measure zero. With that refinement (which we typically never think about) we can show that $\displaystyle d$ is a metric. This is how to get the proof started. Last edited by Maschke; March 2nd, 2017 at 12:07 PM. 
March 2nd, 2017, 03:38 PM  #4 
Global Moderator Joined: May 2007 Posts: 6,454 Thanks: 567  
March 3rd, 2017, 09:27 AM  #5 
Senior Member Joined: Mar 2015 From: New Jersey Posts: 1,297 Thanks: 94 
Function Space: Function considered as infinite dimensional vector. What is size (Norm) of a vector? What is distance d between vectors? In Euclidean space, N=a and d=ba. Generalization to function space, L$\displaystyle ^{p}$[a,b]: Definition, pintegrable: Lim $\displaystyle \Sigma f(x_{i})^{p}\Delta x_{i}$ exists and denoted by $\displaystyle \int f(x)^{p}dx$ $\displaystyle \int$ is Riemannian, or other if dx is d$\displaystyle \mu$, where $\displaystyle \mu$ is a defined measure. Norm: N = f = $\displaystyle \left [ \int f(x)^{p}dx \right ]^{ 1/p}$ Application: Express arbitrary functions in terms of basis functions. Fourier series for example. Which brings us to OP: Given: pintegrable (Riemannian) function space. Does $\displaystyle f_{1}f_{2}$ satisfy requirements of distance function? It does if it satisfies triangle inequality: $\displaystyle f_{1}f_{3}\leq f_{1}f_{2}+f_{2}f_{3}$: https://press.princeton.edu/chapters/s9627.pdf Ttheorem 1.2 

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