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 ...

Read More »## 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 ...

Read More »## 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 ...

Read More »## 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 ...

Read More »## Suppose K and C are subsets of a topological vector space X, K is compact, C is closed, and K n C = ∅. Then 0 has a neighborhood of V such that (K + V) n (C + V) = ∅.

Proof: If K = ∅, then K + V = ∅, and the conclusion of the theorem is obvious. We therefore assume that K # ∅, and consider a point x ∈ K. Then x ∉ C ⟹ x ∈ CC. Since C is closed, ∃ a neighborhood W of ...

Read More »## Useful definitions for functional analysis

Useful definitions for functional analysis Normed spaces: A vector space X is said to be a normed space if to every x∊ X there is associated a nonnegative real number l l x ll, called the norm of x, in such a way that (a) || x + Y || ...

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