# Lebesgue Space is Vector Space

## Theorem

Let $\struct {X, \Sigma, \mu}$ be a measure space and let $p \in \closedint 1 \infty$.

Let $\map {\LL^p} {X, \Sigma, \mu}$ be Lebesgue $p$-space on $\struct {X, \Sigma, \mu}$.

Then $\map {\LL^p} {X, \Sigma, \mu}$ is a vector subspace of $\map \MM {X, \Sigma, \R}$, the space of real-valued $\Sigma$-measurable functions on $X$.

In particular, it is a vector space over $\R$.

## Proof

$\map \MM {X, \Sigma, \R}$ forms a vector space with pointwise addition and pointwise scalar multiplication.

From construction, we have:

$\map {\LL^p} {X, \Sigma, \mu} \subseteq \map \MM {X, \Sigma, \R}$

Since $0 \in \map {\LL^p} {X, \Sigma, \mu}$, we have $\map {\LL^p} {X, \Sigma, \mu} \ne \O$.

From the One-Step Vector Subspace Test, it suffices to show that for each $f, g \in \map {\LL^p} {X, \Sigma, \mu}$ and $\lambda \in \R$ we have:

$f + \lambda g \in \map {\LL^p} {X, \Sigma, \mu}$

### Case 1: $1 \le p < \infty$

Since:

$\ds \paren {\int \size g^p \rd \mu}^{1/p} < \infty$

we have:

 $\ds \paren {\int \size {\lambda g}^p \rd \mu}^{1/p}$ $=$ $\ds \paren {\int \size \lambda^p \size g^p \rd \mu}^{1/p}$ $\ds$ $=$ $\ds \paren {\size \lambda^p \int \size g^p \rd \mu}^{1/p}$ Integral of Positive Measurable Function is Positive Homogeneous $\ds$ $=$ $\ds \size \lambda \paren {\int \size g^p \rd \mu}^{1/p}$ $\ds$ $<$ $\ds \infty$

From Minkowski's Inequality on Lebesgue Space, we have:

$f + \lambda g \in \map {\LL^p} {X, \Sigma, \mu}$

as required.

$\Box$

### Case 2: $p = \infty$

From $g \in \map {\LL^\infty} {X, \Sigma, \mu}$, we have:

$\map \mu {\set {x \in X : \size {\map g x} \ge c} } = 0$ for some $c > 0$.

Then we have:

$\map \mu {\set {x \in X : \size {\map g x} \ge c} } = \map \mu {\set {x \in X : \size {\lambda \map g x} \ge \size \lambda c} } = 0$

So $\lambda g \in \map {\LL^\infty} {X, \Sigma, \mu}$.

Then, from Minkowski's Inequality on Lebesgue Space, we have:

$f + \lambda g \in \map {\LL^\infty} {X, \Sigma, \mu}$

as required.

$\blacksquare$