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In the book "Calculus of Variations" by Gelfand and Fomin, (continuous) linear functionals are defined as follow in the image:definition of linear functional

The fact that continuous is in parentheses seems to suggest that they implicitly assume that linear functionals they consider are continuous. This seems also confirmed by the four examples they give just after this definition, which are all four continuous linear functionals.

In the book, they also define the variation or differential of a functional $J$ by the linear functional $\phi$ such that $\Delta J[h] = J[y+h]-J[y] = \phi[h] + \epsilon \Vert h \Vert$, where $\epsilon$ converges to $0$ as $\Vert h \Vert$ converges to $0$. I am not sure if here they assume $\phi$ continuous (in which case the definition of the variation coincides with the Fréchet differential) or if the continuity is not required?

(I saw that a similar question has already been posted on the forum but the answers were not clear)

dylan
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  • At a guess they intend to mimic the Fréchet definition. But you can check yourself if continuity is or isn’t needed in the proofs they demonstrate – FShrike Apr 28 '23 at 19:02
  • @FShrike I tried to find evidences of continuity in the proofs of theorems but I did not succeed. However, they also define the second variation as the quadratic functional $\phi_2$ such that $\Delta J[h] = \phi_1[h] + \phi_2[h] + \epsilon \Vert h \Vert ^2$, where $\phi_1$ is the first variation. Again they did not explicitly required $\phi_2$ to be continuous. It is quite easy to see that $\phi_1$ is indeed the first variation if $\phi_2$ is continuous, if $\phi_2$ is not continuous I do not know how to prove it. This makes me thing that the continuity is implicit but I am not 100% sure. – dylan Apr 28 '23 at 19:26

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