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Showing that a function is in [itex]L^2(\mathbb{R}^2)[/itex]

  1. Mar 31, 2013 #1
    1. The problem statement, all variables and given/known data
    Suppose [itex]f\in L^2(\mathbb{R}^2)[/itex]. Is [itex]f+c\in L^2(\mathbb{R}^2)[/itex] where c is a constant?

    2. Relevant equations
    [itex]f\in L^2(\mathbb{R}^2)[/itex] if [itex]||f||_2<∞[/itex].


    3. The attempt at a solution
    I think the answer is no because [itex]∫_{\mathbb{R}^2}{c^2}dx=∞[/itex]. However, I am still unsure. Any guidance is appreciated.
     
  2. jcsd
  3. Apr 1, 2013 #2

    jbunniii

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    You are correct. For a simple counterexample, take ##f(x) = 0## for all ##x##. Then ##f \in L^2##, but if ##c## is any nonzero constant, then ##f + c \not\in L^2##, for the reason you indicated.
     
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