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I'm reading the Wikipedia article on distributions. They say that there's a distribution [itex]T_f[/itex] for each function f, defined by
[tex]\langle T_f,\phi\rangle=\int f\phi dx[/tex]
and that there's a distribution [itex]T_\mu[/itex] for each Radon measure [itex]\mu[/itex], defined by
[tex]\langle T_\mu,\phi\rangle\int \phi d\mu[/tex]
They define the delta distribution by
[tex]\langle\delta,\phi\rangle=\phi(0)[/tex]
I feel that there's one thing missing in all of this. I don't see an explanation of the expression
[tex]\int \delta(x)f(x)dx[/tex]
What does this integral mean? Is there a definition of the integral of a distribution that applies here, or should this just be interpreted as a "code" representing the expression [itex]\langle\delta,\phi\rangle[/itex]? (This would mean that the expression has nothing to do with integrals at all, but is written as if it were an integral just to make it look like the expression involving f above).
Also, can someone tell me why you can define a topology on the test function space by defining limits of test functions? (If you answer by referencing a definition or a theorem in a book, make sure it's either "Principles of mathematical analysis" by Walter Rudin or "Foundations of modern analysis" by Avner Friedman, because those are the only ones I've got
)
[tex]\langle T_f,\phi\rangle=\int f\phi dx[/tex]
and that there's a distribution [itex]T_\mu[/itex] for each Radon measure [itex]\mu[/itex], defined by
[tex]\langle T_\mu,\phi\rangle\int \phi d\mu[/tex]
They define the delta distribution by
[tex]\langle\delta,\phi\rangle=\phi(0)[/tex]
I feel that there's one thing missing in all of this. I don't see an explanation of the expression
[tex]\int \delta(x)f(x)dx[/tex]
What does this integral mean? Is there a definition of the integral of a distribution that applies here, or should this just be interpreted as a "code" representing the expression [itex]\langle\delta,\phi\rangle[/itex]? (This would mean that the expression has nothing to do with integrals at all, but is written as if it were an integral just to make it look like the expression involving f above).
Also, can someone tell me why you can define a topology on the test function space by defining limits of test functions? (If you answer by referencing a definition or a theorem in a book, make sure it's either "Principles of mathematical analysis" by Walter Rudin or "Foundations of modern analysis" by Avner Friedman, because those are the only ones I've got