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  61. begin{document}
  62. title{A study}
  63. author{Hiren Garai}
  64. date{}
  65. begin{frame}
  66. begin{block}{begin{center} huge textbf{A study}
  67.  end{center}}
  68. end{block}
  69. begin{center}
  70. textcolor{blue}{Hiren Garai} \
  71. textcolor{black}{small{Department of Mathematics
  72. }}
  73. end{center}
  74. end{frame}
  75. section{Introduction}
  76. begin{frame} frametitle{Introduction}
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  79. Functional Analysis is mainly concerned with the structure of infinite dimensional vector spaces and transformations, which are frequently called operators between such spaces.
  80. In the sixities, 2-metric spaces were introduced by G"ahler cite{Ga 1}, cite{Ga 2}.
  81. 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:
  82. begin{itemize}
  83. item[(D1)] ; For distinct points $x, y,in X$, there is $z in X$, such that $d(x,y,z) neq 0.$
  84. item[(D2)] ; $d(x,y,z) = 0 $ if two of the triple $x,y,z in X$ are equal.
  85. item[(D3)] ; $d(x,y,z) = d(x,z,y) = cdots $ (symmetry in all three variables).
  86. 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. $
  87. end{itemize}
  88. is called a $2$-metric, on $X.$ The set $X$ equipped with such a 2-metric is called a textit{$2$-metric space.}
  89.  end{Definition}
  90. end{Document}
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