# Showing that a function is in $L^2(\mathbb{R}^2)$

1. Mar 31, 2013

### lmedin02

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

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

3. The attempt at a solution
I think the answer is no because $∫_{\mathbb{R}^2}{c^2}dx=∞$. However, I am still unsure. Any guidance is appreciated.

2. Apr 1, 2013

### jbunniii

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.