This book covers some aspects of the theory of Topological algebras including Frechet algebras and Q-algebras. It is devided into three chapters. In the first chapter, the basic concepts necessary for the understanding of the subject matter are introduced. In chapter 2 we show that a continuous function on a hemicompact k-space is of exponential type under certain conditions. Using this result, we generalize the Arens-Royden Theorem to Frechet algebras and obtain interesting results on the density of invertible elements in uniform Frechet algebras. We also find a topological condition to characterize hemicompact k-spaces on which every continuous function has square root. In chapter 3 we show that in a topological algebra, the upper semicontinuity of the spectrum function, the upper semicontinuity of the spectral radius function, the continuity of the spectral radius function at zero, and being a Q-algebra, are all equivalent. Then it is shown that every homomorphism from a Q-algebra with a complete metrizable topology onto a semisimple Frechet algebra is automatically continuous. Also we get some results on the continuity of homomorphisms between regular Frechet algebras

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M. N. Tavani is Ph.D.(2007)from Islamic Azad University, Science and Research branch and assistant professor in Islamic Azad university, Islamshahr branch. She is a researcher in the area of Functional Analysis specializing in Frechet and Banach Algebrs. She has published several research papers in Iranian and foreign journals.