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Recently I came across a definition to which I'd never given much thought. I was reading through Gelfand and Fomin's "Calculus of variations" and I read the part about weak and strong extrema, and I really can't manage to wrap my head around these definitions. They can be found in the wiki article (basically a copy-paste of the book):

http://en.wikipedia.org/wiki/Calculus_of_variations#Extrema

My immediate thought is that these definitions must be inverted. I mean, if you have:

[tex] \| f-f_0 \|_1 < \delta [/tex]

For certain f's, then, a fortiori you have:

[tex]\| f-f_0\|_0 < \delta [/tex]

For all these f's. So this means that the extremum defined with the 1st order norm should be stronger!

At some point I found a definition that was the exact opposite of the one given in wiki (in some google book), but I can't find it anymore. Maybe I should just sleep on it, but I'd still like your input. While we're at it, can you recommend a good, modern book on the calculus of variations? I find that most books on the subject used as references are pretty dated and often not very clear.

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# Extrema of functionals

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