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 || ...

Read More »## 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 ...

Read More »## 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. ...

Read More »## 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 ...

Read More »## Conditions of congruence of triangles

Definition of congruence: If two figures exactly cover each other then they are called congruent. Definition of congruence of triangles: If a triangle when placed on another, exactly covers the other, the triangles are congruent. Two triangles will be congruence if – Side Angle Side( SAS): If two pairs ...

Read More »## SHORT BIOGRAPHY OF EUCLID

EUCLID Born: c. 365 BC Birth place: Alexandria, Egypt Education:Plato’s Academy Athens Greece Occupation: Mathematician, Educator Nationality: Ancient Greace Famous as: Mathematician – Euclidean Geometry, Euclid’s Elements and Euclidean Algorithm Executive summary: Father of geometry Died:c.275 BC Died At Age -70 Location of death: unspecified Childhood: Very few little ...

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