Could someone help explain why we get inclusion after the inequality involving norms at the end of the proof?
Why does $\|x\|_q\leq\|x\|_p$ imply that $\ell_p\subset\ell_q$?
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real-analysis
functional-analysis
lp-spaces
1 Answers
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Because the proof showed that given an arbitrary element $x$ of $\ell_p$, we have that $x\in\ell_q$, which is the definition of inclusion
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1Might be helpful to add why this is a consequence of the fact that $|x|_q\le |x|_p.$ If the RHS is finite (i.e. $x\in l^p$), so must be the LHS (so $x\in l^q$). – 2017-01-22