Abstract: Let X be a non-Archimedean Banach algebra with identity e, ( || e  || = 1) , over non-trivially 

valued field k, we assume that || X || = | k |. For each x  X and a maximal ideal M of X let x(M) denote 

the coset of X/M to which x belongs. X/M is a field which is, in general, an extension of k. We assume 

when necessary X/M to be isomorphic to k for every maximal ideal M of X. The structure of a 

non-Archimedean Banach algebra has been investigated and related theorems are given. Several basic 

differences between this setting and the classical situation are pointed out and examples to show these 

deviations have been given. Let s(x) denote the spectrum of  x  X and  r_s (x) the spectral radius. It 

has been observed that s(x) may be empty: non-trivial examples are given to justify this conclusion. 

Moreover a criteria for the invertibility of elements of X is given. Several theorems concerning spectrum 

and spectral radius which are similar in nature to the classical results are proved. Conditions are given to 

guarantee the equality of spectral radius and the norm. The one-to-one correspondence between the 

maximal ideals of X and the maximal ideals of X is proved under certain basic conditions. The 

compactness of the space of maximal ideals is Gelfand topology is guaranteed if the ground field is locally 

compact. The notion of unique norm topology in this setting has been investigated. Let || || ₁ , || || ₂ be two 

non-Archimedean norms on X and suppose characteristic of  k # 2. For s  X, let 

(s) = inf _{x  X} ( || x ||₁+ || s - x ||₂ ); is called the separating function for the two norms, several 

properties of (s) are given. It has been shown that (s)  is a non-Archimedean psuedo norm on X, s is 

called a separating element if (s) =  0 . The set of separating elements S is an ideal of X called the 

separating ideal. It has been shown that two norms on X are equivalent iff S = { 0 }. Results include: 

Theorem. If X is simple algebra , then X has a unique norm topology. 

Theorem: If X is commutative semisimple algebra, then X has a unique topology. 

Theorem. The separating ideal is contained in the radical of X where X is commutative algebra. 

               Let E be a non-Archimedean normed space over non-trivially valued field  k. The notion of 

best approximation has been investigated. Let V be a closed subspace of E; x  V is said to be a best 

approximation of x  E , x  V if || x - x || = inf _gV  || x - g ||. It has been shown that if best approximation 

exists, then it is never unique. Concerning the existence the main result is: 

               Theorem. If for every closed proper subspace V of E, there exists at least one element x₀E , 

x₀ V having a best approximation in V, then for any closed subspace V of E every element in E has a 

best approximation in V. This gives an answer to a question posed by F. Monna concerning the existence 

of best approximation. That the condition of Monna for the existence of best approximation is not 

sufficient has been shown by a counterexample. Moreover it has been shown by a counterexample that 

the set of elements of E which have best approximation in a given subspace V of E need not be, in 

general, a space. 
 
                Application of our results to non-Archimedean function theory has been given. Results include: 

The algebra T_n of strictly convergent power series is anon-Archimedean Banach algebra which is not 

regular in the sense that the Zariski's topology does not coincide with the Gelfand topology. Moreover 

we give a non-trivially valued field where the norm satisfies the strong product rule, but it is not 

isomorphic to the ground field. We conclude the thesis with comments on some problems prevailing to 

our investigation.