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

[lmedin02]

Homework Statement

Suppose $f\in L^2(\mathbb{R}^2)$. Is $f+c\in L^2(\mathbb{R}^2)$ where c is a constant?

Homework Equations

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

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.

 

[jbunniii]

Homework Statement

Suppose $f\in L^2(\mathbb{R}^2)$. Is $f+c\in L^2(\mathbb{R}^2)$ where c is a constant?

Homework Equations

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

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.

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.