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Is this true?

$L^{1} [0, 1] \subseteq L^{2} [0, 1]$

No — it is false.

The counterexample

$f (x) = 1/ x$

$\int f = 2 < \infty$ but $\int f^{2} = \int$ dx $/ x = \infty$ . On a finite measure space the inclusion runs the other way.

The kind of mistake this is

Converse assumed

The theorem runs one way. You used it in the other.

Drill statements like this

Others that fail the same way

From Lebesgue Measure and Integration › L^p spaces essentials

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