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## Shah Alam

I am a student of mathematics department of Begum Rokeya University, Rangpur. I want to provide all kinds of solution of mathematics problems by this web site.

## 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 ...

## 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 ...

## 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 ||  ...

## Short biography of Isaac Newton

PDF FILE OF SHORT BIOGRAPHY OF ISAAC NEWTON                  Newton   Birth and family Isaac Newton was born on January 4, 1643 in the tiny village of Woolsthorpe-by-Colsterworth, Lincolnshire, England. His father, whose name was also Isaac Newton, was a farmer who died before Isaac Junior was born. His mother, Hannah ...

## PDF| Solution of Exercise- 8.1(Circle) of class 9 and 10

PDF| Solution of Exercise: 8.1(Circle) of class 9 and 10 PDF file of Chapter 8.1: CLICK HERE Exercise 8.1 _Class – 9 and 10_   Problem – 1 :  Prove that if two chords of a circle bisect each other, their point of intersection is the centre of the circle. ...

## Biography of Pythagoras

Born: Approximately 569 BC Birth place: Samos Greece Founder of a philosophical and religious school Died: Approximately 500 – 475 BC Died At Age – About 70 – 95 Location of death: Metapontum Italy     Pythagoras is repeatedly referred to as the first pure mathematician. He was born on ...