continuous embedding in $L^p$ spaces

Question

Let $(\Omega,M,\mu)$ be a complete measure space.

Is it true that $L^1(Ω,M,μ) ∩ L^∞(Ω,M,μ)$ is continuously embedded in $L^p(Ω,M,μ)$ for all $p∈[1,∞]$?

I think yes:

Let $I\colon X \to Y$, $Ix = x$ where $X = L^1(Ω,M,μ) ∩ L^∞(Ω,M,μ)$, $Y = L^p(Ω,M,μ)$ for any $p$.

I have to prove that $I$ is bounded and this is trivial thanks to the inequality

$$‖‖_ ⩽ (μ(Ω))^{1/p} ‖‖_∞$$

Am I missing something?

Answer 1

On the real line with Lebesgue measure let $f_n=\frac 1 n \chi_{(n,n+n^2)}$. Then $f_n \in L^{1}\cap L^{\infty}$ and $f_n \to 0$ in $L^{\infty}$ norm. But $f_n$ does not converge in $L^{2}$. So your map is not continuous when $X$ is given $\|f\|_{\infty}$ norm.