Pure maths with Usama

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Assalam alaikum My channel is about pure mathematics The main focus of this channel is on three subjects of pure mathematics 1) Group Theory 2) Topology 3) S...

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Any reparametrization of a regular curve is regularUnit speed curve is regular

ReparametrizationAny regular curve admits a reparametrization in terms of arc length

Arc length function for a parametrized curve speed of a curveUnit speed curve

Tangent vector of a parametrized curveFunction of class n and class infinitySmooth function

Differential geometryParametrization Parameterized curve Level curveAstroidIMM-LUMS

Automorphisms of cyclic group of order nZnCnAut(Cp) Automorphisms of Z4, Z5 and Z6

The dual space of l^p space is l^q space where p and q are conjugate indexSequence space

Isometric isomorphismDual space of R^n is R^nDual space of l1 is l infinity Algebraic dual space

Space of all bounded linear operatorsDual spaceB(X,Y) is a Banach space if Y is Banach space

Dimension of Algebraic dual space A finite dimensional vector space is algebraically reflexive

Dual basis for Algebraic dual space Linear functional Kronecker Delta Examples

Every linear operator on a finite dimensional vector spaces represent a matrix and vise versaMatrix

Let S be a closed subspace of a Banach space N.Then NS is also a Banach spaceFunctional Analysis

Quotient spaceHyperplaneIf S is closed subspace of a normed space then NS is also a normed space

Convex HullConvex Hull is collection of all Linear Convex combinations Closure of a convex set

For any normed space, the open and closed balls are convex Convex Set

Convex Setimage of a convex Set under linear transformation is convexEvery subspace is convex

formula to find total no of permutations conjugate with α . Conjugacy classes of Symmetric group Sn

Let Sn be a symmetric group. Total no of cycles of length m in Sn is n!m(n-m)! i.e. npmm

Baire's category theoremEvery Countable intersection of dense open sets is dense

Order structure of S4 and S5Formula to find total no of elements of order r in S4 and S5

Baire's Category theoremA complete metric space is of second categoryIsolated point

First category and second categoryQ is of first categoryZ is of second category as a subspace

Nowhere dense subsetClosure of A has empty interiorcomplement of closure of A is dense in X

Quran recitation last two ayat of Surat-Al-Baqara

C[a,b] space of all continuous functions is complete C[0,1] is incomplete metric space

The sequence space lp is complete

The space R^n(or C^n) is completeThe space ℓ∞ of all bounded sequences is complete

The space R with the usual metric is a complete space

The space C of complex numbers with the usual metric is a complete metric space