Suppose we have a Hilbert space $X$ and its dual $X^*$. Given a dual pairing $$_{X^*}\langle x,y\rangle_X,$$ does there exist a sort of Cauchy-Schwarz inequality so that $|\langle x, y\rangle|\leq ||x||_{X^*}||y||_{X}$?
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A Hilbert space is isomorphic ti its dual/ – DanielWainfleet Apr 14 '16 at 00:30
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Yes, since $|⟨ x,y ⟩ | = |x(y)| \leq \sup_{‖z‖_X=1} |x(z)|‖y‖ = ‖x‖_{X^*}‖y‖_X $.
Calvin Khor
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Do we actually need $X$ to be a Hilbert space or does this inequality hold for general Banach spaces? – Alvo Jun 22 '19 at 07:16
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1@Alvo Yes, this is nothing but the definition of the norm of an linear functional. – Calvin Khor Jun 22 '19 at 09:08
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You mean "Yes, it does hold for general Banach spaces"? Sorry, just want to be sure... – Alvo Jun 22 '19 at 09:32
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1If you really want to be sure, you should consult the definition of the norm of a linear functional instead of trusting something a stranger wrote :) @Alvo – Calvin Khor Jun 22 '19 at 09:33
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Does an inequality of such sort still hold for non-linear functionals ? – Rebellos Jan 16 '20 at 17:12
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@Rebellos I have no clue how to answer such a vague question, much less in a comment. Ask a new question? – Calvin Khor Jan 17 '20 at 01:18