Triangle Inequality for Series/Lebesgue Spaces

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Theorem

Let $\left({X, \Sigma, \mu}\right)$ be a measure space.

Let $p \in \R$, $p \ge 1$.

Let $\left({f_n}\right)_{n \in \N} \in \mathcal{L}^p \left({\mu}\right)$ be a sequence of $p$-integrable functions, i.e., a sequence in Lebesgue $p$-space.


Suppose that for all $n \in \N$, $f_n \ge 0$ holds pointwise.

Then:

$\displaystyle \left\Vert{\sum_{n \mathop = 1}^\infty f_n}\right\Vert_p \le \sum_{n \mathop = 1}^\infty \left\Vert{f_n}\right\Vert_p$

where $\left\Vert{\cdot}\right\Vert_p$ denotes the $p$-seminorm.


Proof


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