Let $f: E \to F$ be a $\mathcal C^1$-map between locally convex spaces. Is the following correct $$ \mathrm{d}f(x)(h) = \langle f'(x),h\rangle\:\:? $$
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2You might want to explain your notation. What is $f'(x)$ for you? – Severin Schraven Aug 24 '19 at 20:18
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In case you are asking whether the Gateaux derivative at $x$ of $f$ in direction $h$ is equal to the Frechet differential of $f$ at $x$ applied to $h$; yes, that is true. – Severin Schraven Aug 24 '19 at 20:25
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Thanks, Severin. I meant that. Is it true for any topology on the dual space? – Richard Kim Aug 24 '19 at 20:40
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1What do you mean by the dual space? – Severin Schraven Aug 24 '19 at 20:42
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the differential df( x) belongs to the dual space E', and <,> is the duality between E' and E. I know this is true for the Banach case but I was as not sure about locally convex space because there different topologies on E'. – Richard Kim Aug 24 '19 at 20:51
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You can google On Differentials in Locally Convex Spaces, then it should be one of the first theorems. – Severin Schraven Aug 24 '19 at 20:55
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I couldn't find it, I saw just for Banach spaces – Richard Kim Aug 24 '19 at 20:57
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What do you mean? On Differentials in Locally Convex Spaces is the name of the paper by Falb and Jacobs. Just google it. – Severin Schraven Aug 24 '19 at 21:01
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I just read that paper but there is nothing about that. – Richard Kim Aug 24 '19 at 21:07
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I am confused, doesn't Theorem 1 exactly say what you want? – Severin Schraven Aug 24 '19 at 21:08
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Theorem 1 says that the strong differentibility implies weak differentiability but it doesn't mention anything about duality – Richard Kim Aug 24 '19 at 21:12
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It says $$ d_w f(x,h) = df(x,h) $$ which is what you want. The LHS is the directional derivative of $f$ at $x$ in direction $h$ and the RHS is the derivative of $f$ at $x$ applied to $h$. – Severin Schraven Aug 24 '19 at 21:15