Minkowski's Inequality and the Seminormed Space of Power-Integrable Functions

In the setting of Measure Spaces and the Lebesgue Integral: Standing Notation, let $(X,\mathcal{F},\mu)$ be a measure space, let $p$ be a real number with $1\le p$, and write $\mathcal{L}^{p}$ for the set of $p$-integrable functions on $(X,\mathcal{F},\mu)$ and $\lVert\cdot\rVert_{p}$ for the $p$-seminorm. Then the following hold.

1. (Minkowski's inequality)¶ Let $f,g\in\mathcal{L}^{p}$. Then the pointwise sum $f+g$ belongs to $\mathcal{L}^{p}$ and

2. (A seminormed vector space)¶ $\mathcal{L}^{p}$ is a linear subspace of the real vector space of all real-valued maps on $X$, and is therefore itself a vector space over $\mathbb{R}$ under the pointwise operations, with zero vector the map taking the value $0$ at every point. For all $f,g\in\mathcal{L}^{p}$ and every real $c$ the $p$-seminorm satisfies

3. (Finite sums)¶ Let $n\in\mathbb{N}$ and let $f_{k}\in\mathcal{L}^{p}$ for each $k\in[n]$. Then the finite sum $\sum_{k=1}^{n}f_{k}$, formed in the vector space of clause 2, belongs to $\mathcal{L}^{p}$ and