Is the following statement true? L¹[0,1] ⊆ L²[0,1]

No. f(x) = 1/√x ∫f = 2 < ∞ but ∫f² = ∫dx/x = ∞. On a finite measure space the inclusion runs the other way.

Counterexample

Is this true?

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

No — it is false.

The counterexample

$f(x) = 1/\sqrt{x}$

Why it fails

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

Keep practicing

Converse assumed

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

Published 23 Aug 2026. Found a mistake? Tell us — corrections are the fastest thing we act on.