Functional Analysis Answers

prove that a finite partially ordered set A has at least one maximal element

Please, this is urgent , solve it as soon as possible.

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

Let TES^1. Prove that for all multi-indices a, (D^(a)T)^ = x^(a)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.