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Subadditive Function

Posted: Thu Jul 14, 2016 12:52 pm
by Tsakanikas Nickos
Let \( \displaystyle X \) be a normed space and let \( \displaystyle p: X \longrightarrow \mathbb{R} \) be a subadditive function. Show that
  • If \( \displaystyle p(0)=0 \) and \( \displaystyle p \) is continuous at \( 0 \), then \( \displaystyle p \) is continuous at every \( \displaystyle x \in X \)
  • If \( \displaystyle p \) is non negative outside a sphere \( \displaystyle \left\{ x \in X \;\big| \; \|x\|=r \right\} \), then \( \displaystyle p \) is non negative on \( \displaystyle X \)