\documentclass[a4,12pt]{amsart} \usepackage{amsmath, amsthm, amscd, amsfonts, amssymb, graphicx, color,float,pgf,tikz} \usepackage{amssymb,fontenc} \usepackage{latexsym,wasysym,mathrsfs} \usepackage{hyperref} \usetikzlibrary{arrows} \linespread{1.5} \makeatletter \oddsidemargin.9375in \evensidemargin \oddsidemargin \marginparwidth1.9375in \makeatother \def\affilfont{\normalfont\fontsize{10}{12}\selectfont\centering}%\itshape%SHK 24/2 \def\affil#1{\par\vskip12pt{\affilfont#1\par}\vskip15pt} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{conjecture}[theorem]{Conjecture} \newtheorem{note}[theorem]{Note} \theoremstyle{definition} \newtheorem{definition}[theorem]{Definition} \newtheorem{example}[theorem]{Example} \newtheorem{xca}[theorem]{Exercise} \theoremstyle{remark} \newtheorem{remark}[theorem]{Remark} \numberwithin{equation}{section} \newcommand{\R}{{\mathbb R}} \newcommand{\N}{{\mathbb N}} \newcommand{\Z}{{\mathbb Z}} \newcommand{\D}{{\mathbb D}} %%%%%%%%%%%%%%Landausymbole%%%%%%%%%%%%%%%%%%%%%%%% \newcommand{\OO}{\mathcal{O}} \newcommand{\oo}{o} %%%%%%%%%%%%%%%%%%ABBRS%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \newcommand{\nn}{\nonumber} \newcommand{\be}{\begin{equation}} \newcommand{\ee}{\end{equation}} \newcommand{\ul}{\underline} \newcommand{\ol}{\overline} \newcommand{\ti}{\tilde} \newcommand{\spr}[2]{\langle #1 , #2 \rangle} \newcommand{\id}{{\rm 1\hspace{-0.6ex}l}} \newcommand{\E}{\mathrm{e}} \newcommand{\I}{\mathrm{i}} \newcommand{\sgn}{\mathrm{sgn}} \newcommand{\tr}{\mathrm{tr}} \newcommand{\im}{\mathrm{Im}} \newcommand{\re}{\mathrm{Re}} \newcommand{\dom}[1]{\mathrm{dom}\left(#1\right)} \newcommand{\mul}[1]{\mathrm{mul}\left(#1\right)} \newcommand{\col}{\mathrm{col}} \newcommand{\fdom}{\mathfrak{Q}} \newcommand{\cR}{\mathcal{R}} \newcommand{\Nk}{\mathbf{N}_\kappa} \newcommand{\Nak}{\mathbf{N}_\kappa^a} \DeclareMathOperator{\ran}{ran} \DeclareMathOperator{\res}{Res} \DeclareMathOperator{\rang}{rank} \DeclareMathOperator{\conv}{conv} \DeclareMathOperator{\reg}{r} %%%%%%%%%%%%%%%GREEK%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \newcommand{\eps}{\varepsilon} \newcommand{\vphi}{\varphi} \newcommand{\sig}{\sigma} \newcommand{\lam}{\lambda} \newcommand{\gam}{\gamma} \newcommand{\om}{\omega} \newcommand{\al}{\alpha} \newcommand{\ta}{\theta} %%%%%%%%%%%%%%%%%%%%%%%%NUMBERING%%%%%%%%%%%%%%%%%%%%%%%% \renewcommand{\labelenumi}{(\roman{enumi})} \renewcommand{\theenumi}{\roman{enumi}} \numberwithin{equation}{section} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{document} \noindent {\tiny Name of Journal Vol. $\cdots$ No. $\cdots$(2015), $\cdots$-$\cdots$\\ %\url{http://scma.maragheh.ac.ir}} \\[0.50in] %-------------------------------------------------------------- \title[Fuzzy Bases and Fuzzy frame in Fuzzy Hilbert spaces]{Fuzzy Bases and Fuzzy frame in Fuzzy Hilbert spaces} %-------------------------------------------------------------- \author[B. Daraby]{Bayaz Daraby$^1$$^{*}$} \address{ $^{1}$ Associated Professor in Mathematical Analysis, Department of mathematics, University of Maragheh, Amirkabir highway, P.O.Box: 55181-83111, Maragheh, Iran.} \email{bdaraby\@ maragheh.ac.ir} \author[A. Rostami]{Ali Rostami$^2$} \address{ $^{2}$ Phd student from the University of Maragheh.} \email{aliros58 \@ yahoo.com} %--------------------------------------------------------------- \subjclass[2010]{03E72, 15A63.} %--------------------------------------------------------------- \keywords{Fuzzy inner product, Fuzzy frame, Fuzzy inner product, Fuzzy base.\\ %\indent Received: dd mmmm yyyy, Accepted: dd mmmm yyyy. %\\ \indent $^{*}$ Corresponding author} \maketitle %\vspace{0.1in} \hrule width \hsize \kern 1mm %-------------------------------------------------------------- \begin{abstract} In this paper, first the concept of fuzzy inner product is presented by PINAKI MAJUMDAR and S.K. SAMANTA expressed and the concept of fuzzy bases in fuzzy Hilbert spaces introduced. The some proposition and an important theorem about the fuzzy bases are proved. Then the concept of fuzzy frames in fuzzy Hilbert spaces will be introduced. Finally, a major theorem that is relationship between the fuzzy bases and the fuzzy frame proving. \end{abstract} \maketitle \vspace{0.1in} \hrule width \hsize \kern 1mm %---------------------------------------------------------------- \section{Introduction} It was Katsaras \cite{5}, who while studying fuzzy topological vector spaces, was the firs to introduced in 1984, the idea of fuzzy norm on a linear space. Later on many other mathematicians like Felbin \cite{4}, Cheng Mordeson \cite{3}, Bag Samanta \cite{1-2} etc. introduced definition of fuzzy normed linear spaces in dierent approach. Studies on fuzzy inner product spaces are relatively recent and few work have been done in fuzzy inner product spaces. %--------------------------------------------------------------- \section{Strong Convergence} In this section some definitions and preliminary results are given which are used in this paper. \begin{definition}[\cite{1}] Let $U$ be a linear space over $F$ (field of Real/Complex numbers). A fuzzy subsets $N$ of $U \times R$ ($R$ is the set of all real numbers) is called a fuzzy norm on $U$ iff $\forall x,u \in U$ and $c \in F$. \begin{itemize} \item[(N1)] $\forall t \in R$ with $ t \le 0$, $N(x,t)=0$ \item[(N2)] ($\forall t \in R, t>0 , N(x,t) =1$ iff $x=0$ \item[(N3)] $\forall t,s \in R , x, u \in U \quad U(x+u, s+t) \ge \min\{N(x,s) , N(u,t)\}$ \item[(N5)] $N(x)$ is a non-decreasing function of $R$ and $N(x,t) \to a$ as $t \to \infty$ \end{itemize} The pair $(U