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# Functional Analysis

## Baire’s theorem | Functional Analysis

Statement: If X is a complete metric space, the intersection of every countable collection of dense open subsets of X is dense in X. Proof: Suppose v1, v2,… are dense in and open in X. We have to prove that is dense in X. It is suffices to show that ...

## Let X be a topological vector space.If A ⊂ X then A = n (A + V), where V runs through all neighborhoods of O (zero).

Theorem: (x +v) ∩ A ≠ Φ if and only if x ⊂ A – V . Proof: Suppose a ∈ (x + v) ∩ A. Then a ∈ x + V and a ∈ A. But,   a ∈ x + V ⟹ a = x + v ⟹ x ...

## Every topological vector space is a Housdorff space

Proof: Let X be a topological vector space and let x, y ∈, such that x ≠ y. Set K = {x} and C = {y} Then K is a compact and C is a closed and that K ∩ C = Φ. Hence there exist a neighborhood v of ...

## If ℬ is a local base for a topological vector space X, then every member of ℬ contains the closure of some member of ℬ.

Proof: Let W ∈ ℬ. Set K = { 0 } and C = WC. Then C is closed set and K is compact set and C ∩ K = Ф. Hence ∃ a neighborhood V of 0 such that (C+ V) ∩ (K+V) = Ф. Since B is a ...