Let 0 < p < 1 and let X=\f: R R:f continuous, R f p<\. Define f p=( f p)1/p. Which holds?

$|f+g|_p\leq|f|_p+|g|_p$ for all f,g.

Pointwise $f_n\to f$ implies $|f_n|_p\to|f|_p$.

GATE 2020 MA – Question 53

Functional Analysis · Normed and Banach Spaces · 2 marks · Multiple choice

Let 0 < p < 1 and let $X=\{f:\mathbb R\to\mathbb R:f\text{ continuous},\int_{\mathbb R}|f|^p<\infty\}$. Define $|f|_p=(\int|f|^p)^{1/p}$. Which holds?

$|\cdot|_p$ is a norm on X.
$|f+g|_p\leq|f|_p+|g|_p$ for all f,g.
$|f+g|_p^p\leq|f|_p^p+|g|_p^p$ for all f,g.
Pointwise $f_n\to f$ implies $|f_n|_p\to|f|_p$.

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Correct answer: (C) $|f+g|_p^p\leq|f|_p^p+|g|_p^p$ for all f,g.

Explanation

For 0 < p < 1, the scalar inequality $|a+b|^p\leq|a|^p+|b|^p$ holds. Integrating proves C. The ordinary triangle inequality fails: two equal-norm continuous functions with disjoint supports give $|f+g|_p=2^{1/p}|f|_p>2|f|_p$. Pointwise convergence alone also does not control integrals, so neither A, B nor D is generally true.

GATE 2020 MA – Question 53

Explanation

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