Inner Product Space
Two finite dimensional vector spaces over the same field are isomorphic iff they are of the same dimensions
First isomorphism theorem for vector space, homomorphism, kernel and isomorphism of vector space
Dimension theorem of a quotient space: If W be a subspace of a finite dimensional vector space V over ( mathbb{R} ) then (...
Any two right cosets of V/W are either disjoint or identical
First and Second fundamental theorem of calculus
Every continuous function f defined on ([a, b]) is Riemann integrable
A bounded function ( f:[a, b]to mathbb{R} ) is Riemann integrable iff for every ( epsilon>0 ) there exist a partition ( P_epsilon )...
Find the value of $k$, for which $f(x)=begin{cases} frac{sqrt{1+kx}-sqrt{1-kx}}{x}, text{if} -1le x < 0 frac{2x+1}{x-1}, text{if} 0le x < 1 end{cases}$...
A metric space (X, d) is disconnected iff there exists a non-empty proper open & closed subset of X which is both open and...
Any two closed subset of metric space are connected iff they are disjoint
A metric space (X, d) is connected iff every continuous function is constant
Continuous image of connected set is connected
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