Fréchet Algebras with Schauder Bases

Abstract

<p>Let A be a Fréchet algebra with the defining sequence {qn}n≥1 of seminorms and identity e. Let {xᵢ} be a Schauder basis in A. Then each xεA, can be written as: x = ᵢ∑₁αᵢxᵢ, where {αᵢ} is a unique sequence of complex numbers depending upon x. αᵢ's are called coordinate functionals or coefficients. This thesis is concerned with some relations among coefficients, seminorms and the identity, e of A. Further, it is shown that each multiplicative linear functional on A is continuous provided a certain condition is satisfied. Some of the results needed to prove the above results are shown to be true for Fréchet spaces. Finally, a representation theorem for Fréchet * - algebra is given.</p>