L1(E), Lp(E), and L∞(E) Normed Linear Spaces Review

# L1(E), Lp(E), and L∞(E) Normed Linear Spaces Review

We will now review some of the recent material regarding the $L^1(E)$, $L^p(E)$, and $L^{\infty}(E)$ spaces.

• Let $E$ be a measurable set.
• On The L1(E) Normed Linear Space page we defined the set $L^1(E)$ to be the set of all Lebesgue integrable functions on $E$, that is, $f \in L^1(E)$ if $\displaystyle{\int_E |f| < \infty}$. We defined the $1$-norm on $L^1(E)$ to be:
(1)
\begin{align} \quad \| f \|_1 = \int_E |f| \end{align}
• We also showed that $(L^1(E), \| \cdot \|_1)$ is a normed linear space.
• On The Lp(E) Normed Linear Space page we defined the set $L^p(E)$ to be the set of all $p$-Lebesgue integrable functions on $E$, that is, $f \in L^p(E)$ if $\displaystyle{\int_E |f|^p < \infty}$. We defined the $p$-norm on $L^p(E)$ to be:
(2)
\begin{align} \quad \| f \|_p = \left ( \int_E |f|^p \right )^{1/p} \end{align}
• We gave a partial proof showing that $(L^p(E), \| \cdot \|_p)$ is a normed linear space. The only gap that we had was proving that if $f, g \in L^p(E)$ then $\| f + g \|_p \leq \| f \|_p + \| g \|_p$, which is known as Minkowski's inequality for $L^p(E)$, which we will prove later.
• On The L∞(E) Normed Linear Space page we defined the set $L^{\infty}(E)$ to be the set of all essentially bounded functions on $E$, that is, $f \in L^{\infty}(E)$ if there exists an $M > 0$ such that $|f(x)| \leq M$ almost everywhere on $E$. We defined the $\infty$-norm on $L^{\infty}(E)$ to be:
(3)
\begin{align} \quad \| f \|_{\infty} = \inf \{ M : |f(x)| \leq M \: a.e. \: \mathrm{on} \: E \} \end{align}
• We also showed that $(L^{\infty}(E), \| \cdot \|_{\infty})$ is a normed linear space.