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I was looking at this problem: Norm of the operator $Tf=\int_{-1}^0f(t)\ dt-\int_{0}^1f(t)\ dt$ and was confused about the step with the integral inequality:

$$\left|\int_{-1}^0f(t)\ dt-\int_{0}^1f(t)\ dt\right|\\ \leq \int_{-1}^0|f(t)|\ dt+\int_{0}^1|f(t)|\ dt\\$$

Is there a rule for this inequality that I'm missing? In the first equation, if it was addition, I could use the triangle inequality. Here, is it using the assumption that the sum would be greater than the difference?

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user24700
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1 Answers1

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Notice that you can see the first equation as an addition:

$$\begin{align} \left|\int_{-1}^0f(t)\ dt-\int_{0}^1f(t)\ dt\right|&=\left|\int_{-1}^0f(t)\ dt+\left(-\int_{0}^1f(t)\ dt\right)\right|\\\\ &\leq \left|\int_{-1}^0f(t)\ dt\right|+\left|-\int_{0}^1f(t)\ dt\right|\\\\ &= \left|\int_{-1}^0f(t)\ dt\right|+\left|\int_{0}^1f(t)\ dt\right|\\ \\ &\leq \int_{-1}^0|f(t)|\ dt+\int_{0}^1|f(t)|\ dt\\\end{align}$$

Pedro
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