# Triangle Inequality for Series/Lebesgue Spaces

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## Theorem

Let $\struct {X, \Sigma, \mu}$ be a measure space.

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

Let $\sequence {f_n}_{n \mathop \in \N} \in \map {\LL^p} {\mu}$ be a sequence of $p$-integrable functions, that is, a sequence in Lebesgue $p$-space.

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

Then:

- $\displaystyle \norm {\sum_{n \mathop = 1}^\infty f_n}_p \le \sum_{n \mathop = 1}^\infty \norm {f_n}_p$

where $\norm {\, \cdot \,}_p$ denotes the $p$-seminorm.

## Proof

## Sources

- 2005: René L. Schilling:
*Measures, Integrals and Martingales*... (previous) ... (next): $12.6$