Functional Analysis Answers

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Suppose V is finite-dimensional and phi is a linear functional on V. Then

there is a unique vector u in V such that

phi(v)=< v, u> for every v in V.

Show that norm is continuous function.

A metric space M is called a сomplete metric space if

Let TES^1. Prove that for all φES, (T*φ)^ = (2π)^(n/2)(φ-hat)(T-hat).

Let T be a tempered distribution. Then for all multi-indices a, we define ∂^(a)T to be the linear functional on S by (∂^(a)T)(φ) = (-1)^(|a|)T(∂^(a)φ), φES. Prove that ∂^(a)T is a tempered distribution.

find a measer dense subset in r2

If f(x) = f(y) for every bounded linear functional f on a normed space X, show that x = y.

show that the addition of the type dg/dx to the integrand function leaves the euler equation in the same form

Show that p(x)=lim(€n) where x=(€n) belongs to l(infinity) defines a sublinear functional on l(infinity)