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On my textbook (Frigyes Riesz, Bela Sz.-Nagy, functional analysis) there is the following theorem.

Theorem. A necessary and sufficient condition that the functional A, given in a set E of the space C, be extendable to the entire space C so as to define there a linear functional of norm $\leq M$ is that $$\left|\sum_{1}^n c_k A f_k\right|\leq M \left|\left|\sum_{1}^n c_k f_k\right|\right|$$ for every linear combination of elements of E.

The theorem's proof is not very clear for me. Is there any good reference (online, if possible)?

Thanks!

Mark
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  • Continuous means bounded, which is the condition. Then Hahn Banach gives the extension. If it's unbounded, you cannot get an extension which is. – Adam Hughes Jul 13 '14 at 09:56
  • The inequality just ensures that $A$ can be extended to a linear functional on $\operatorname{span} E$ with norm $\leqslant M$. The remaining part is the Hahn-Banach extension theorem. – Daniel Fischer Jul 13 '14 at 09:57
  • Thank you. You can give me a good reference for it? – Mark Jul 16 '14 at 07:33

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