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March 1st, 2017, 11:58 PM   #1
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metric space

How to showed that $\displaystyle \rho(f,g)=\int_{a}^{b}|f(x)-g(x)|^p dx$ defined metric on $\displaystyle L^p[a,b]$?
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March 2nd, 2017, 07:44 AM   #2
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Step 1: Define $\displaystyle L^{p} [a,b]$
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March 2nd, 2017, 11:58 AM   #3
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Quote:
Originally Posted by zylo View Post
Step 1: Define $\displaystyle L^{p} [a,b]$
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.
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Last edited by Maschke; March 2nd, 2017 at 12:07 PM.
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March 2nd, 2017, 03:38 PM   #4
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Quote:
Originally Posted by giokakocha View Post
How to showed that $\displaystyle \rho(f,g)=\int_{a}^{b}|f(x)-g(x)|^p dx$ defined metric on $\displaystyle L^p[a,b]$?
Usually the pth root of the integral defines the metric.
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March 3rd, 2017, 09:27 AM   #5
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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=|b-a|.

Generalization to function space, L$\displaystyle ^{p}$[a,b]:
Definition, p-integrable:
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: p-integrable (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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