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.