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Banach spaces, function spaces, real functions, integral transforms, theory of distributions, measure theory.
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Galerkin approximations for parabolic PDE weak solution, getting a uniform bound
(As usual $V \subset H$ are separable Hilbert spaces)
In a book I read this about existence of the solutions to parabolic PDEs:
the approximate solution $u_n(t)$ solves the equation
$$(u_n', w_ …