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I think distributions are linear and continuous functionals on the set of testfunctions. In a textbook I saw this question:

Let $f$ be a $2\pi$-periodic function with $f(t) = \frac{\pi}{4}|t|$ for $t \in [-\pi, \pi]$, show that it is a distribution!

How could this be, its not even a functional?

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up vote 5 down vote accepted

Locally integrable functions can be indentified to distributions in the sense that if $f$ is locally integrable one associates to it the linear functional $L_f$ defined by $L_f(\phi)=\int f\phi$ for every test function $\phi$.

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