Inequality of the $L_p$ spaces

Question

Assume we have $1\leq p<q<\infty$. How can I show that $L_p(\mathbb{R})\neq L_q(\mathbb{R})$?

I suppose the easiest way would be to show that neither is a subset of the other, but how would I get started on that?

Answer 1

Note $L^q(\mathbb{R})\subset L^p(\mathbb{R})$. Let $$ f(x)=\left\{\begin{array}{ll} \frac{1}{x^\alpha}&\text{ if }x\in(0,1)\\0&\text{ else}\end{array}\right. $$ where $\alpha=\frac{p+q}{2pq}$. Clearly $$ \int_{\mathbb{R}}|f|^pdx=\int_0^1\frac{1}{x^\frac{p+q}{2q}}dx<\infty$$ and $$ \int_{\mathbb{R}}|f|^qdx=\int_0^1\frac{1}{x^\frac{p+q}{2p}}dx=\infty. $$ So $L^q(\mathbb{R})\not=L^p(\mathbb{R})$.