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Is there any main method for finding norm of function in $L_1$ space? For example :

$f(x)$ = $\sin x$ in space $L_1[-\pi,\pi]$

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Welcome to Math.SE! Please edit your question to make it more precise: you say functional but give an example of a function. Which one are you asking about? –  user53153 Jan 6 '13 at 18:16
    
i am sorry,i typed wrong. i meant function –  Javidan Jan 6 '13 at 18:17
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Direct evaluation works for your example. $\|\sin\|_1 = \int_{-\pi}^\pi |\sin x| dx = 2 \int_0^\pi \sin x dx = 4$. –  copper.hat Jan 6 '13 at 18:21
    
then for finding norm of function on $L_1[a,b]$ space, i need to integrate it from a to b ? –  Javidan Jan 6 '13 at 18:31
    
@Javidan: Using the definition of the norm doesn't hurt. In some contexts there may be other methods. Make sure that you integrate the absolute value of the function, i.e., use the definition of the $L_1$ norm. –  Jonas Meyer Jan 6 '13 at 18:37
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1 Answer

up vote 2 down vote accepted

By definition, if $f \in L^p(X)$, then $$\| f \|_p := \left(\int_X |f|^pd\mu\right)^{1/p}.$$

In this specific case case $p=1, X=[-\pi,\pi], f=\sin$, and $\mu$ is the Lebesgue measure we have (as @copper.hat points out) $$ \|\sin||_1 = \int_{-\pi}^{\pi}|\sin(x)|dx = 2\int_0^\pi \sin x dx = 4. $$

That is, to find the norm of a function you literally just plug into the definition.

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