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begin{document}
title{A study}
author{Hiren Garai}
date{}
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textcolor{blue}{Hiren Garai} \
textcolor{black}{small{Department of Mathematics
}}
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section{Introduction}
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Functional Analysis is mainly concerned with the structure of infinite dimensional vector spaces and transformations, which are frequently called operators between such spaces.
In the sixities, 2-metric spaces were introduced by G"ahler cite{Ga 1}, cite{Ga 2}.
begin{Definition} Let $X$ be a nonempty set, and let $R$ be the set of all reals and ${R^+}$ be the set of positive reals. A function $d : X times X times X implies {R^+} cup {{0}}$ satisfying:
begin{itemize}
item[(D1)] ; For distinct points $x, y,in X$, there is $z in X$, such that $d(x,y,z) neq 0.$
item[(D2)] ; $d(x,y,z) = 0 $ if two of the triple $x,y,z in X$ are equal.
item[(D3)] ; $d(x,y,z) = d(x,z,y) = cdots $ (symmetry in all three variables).
item[(D4)] ; $d(x,y,z) leq d(x,y,a) + d(x,a,z) + d(a,y,z), , forall x,y,z,a in X. $
end{itemize}
is called a $2$-metric, on $X.$ The set $X$ equipped with such a 2-metric is called a textit{$2$-metric space.}
 end{Definition}
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