On the set $H^1_0((0,2))$ we put the following norms.
$$\|u\|_a^2= \int_{[0,2]}(u')^2.$$ $$\|u\|_b= \|u\|_\infty.$$ $$\|u\|_c= \|u\|_{L^2}.$$
Is $H^1_0((0,2))$ Banach with any of these norms?
2026-04-12 13:31:18.1776000678
To be or not to be Banach? That is the question.
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Hint: by the Sobolev embedding theorem, if $H_0^2(0,2)$ were complete under either the second or the third norm, then the usual norm would be equivalent to either the $L^\infty$ norm or to the $L^2$ norm. Can you find counterexamples to this?