In Chapter 5, the focus is on analytic functions of several complex variables, particularly holomorphic functions. The chapter begins by recalling the definition of a holomorphic function in a domain \( D \subset \mathbf{C}^n \), which includes the Cauchy-Riemann equations and continuous differentiability. The Hartogs theorem is introduced, stating that if a function is analytic in each component for any values of the remaining components, it is jointly analytic and smooth. The space of all holomorphic functions in \( D \), denoted \( \mathcal{H}(D) \), is a linear subspace of the space of continuous functions \( \mathcal{C}(D) \). The topology on \( \mathcal{H}(D) \) is induced by a system of seminorms defined on compact subsets of \( D \). The concept of \( \mathcal{H}(D) \)-bounded subsets is introduced, where any holomorphic function in \( D \) is bounded in modulus on these subsets. The integral representation for polycircular domains is discussed, where any holomorphic function in a domain \( D \) can be represented by an \( n \)-tuple Cauchy integral within a closed polycircular domain \( P(w; \rho) \). The infinite differentiability of holomorphic functions is established, and a lemma is provided to show that for any compact subset \( K \subset D \), there exists a compact subset \( K' \subset D \) and a polyradius \( \rho \) such that the \( \alpha \)-th derivative of any holomorphic function \( f \in \mathcal{H}(D) \) is bounded by a constant times \( \rho^\alpha \). Finally, it is shown that \( \mathcal{H}(D) \) is a closed subspace of both \( \mathcal{C}(D) \) and \( \mathcal{E}(D) \), making it a Fréchet space with the same LCS structure induced from either \( \mathcal{C}(D) \) or \( \mathcal{E}(D) \).In Chapter 5, the focus is on analytic functions of several complex variables, particularly holomorphic functions. The chapter begins by recalling the definition of a holomorphic function in a domain \( D \subset \mathbf{C}^n \), which includes the Cauchy-Riemann equations and continuous differentiability. The Hartogs theorem is introduced, stating that if a function is analytic in each component for any values of the remaining components, it is jointly analytic and smooth. The space of all holomorphic functions in \( D \), denoted \( \mathcal{H}(D) \), is a linear subspace of the space of continuous functions \( \mathcal{C}(D) \). The topology on \( \mathcal{H}(D) \) is induced by a system of seminorms defined on compact subsets of \( D \). The concept of \( \mathcal{H}(D) \)-bounded subsets is introduced, where any holomorphic function in \( D \) is bounded in modulus on these subsets. The integral representation for polycircular domains is discussed, where any holomorphic function in a domain \( D \) can be represented by an \( n \)-tuple Cauchy integral within a closed polycircular domain \( P(w; \rho) \). The infinite differentiability of holomorphic functions is established, and a lemma is provided to show that for any compact subset \( K \subset D \), there exists a compact subset \( K' \subset D \) and a polyradius \( \rho \) such that the \( \alpha \)-th derivative of any holomorphic function \( f \in \mathcal{H}(D) \) is bounded by a constant times \( \rho^\alpha \). Finally, it is shown that \( \mathcal{H}(D) \) is a closed subspace of both \( \mathcal{C}(D) \) and \( \mathcal{E}(D) \), making it a Fréchet space with the same LCS structure induced from either \( \mathcal{C}(D) \) or \( \mathcal{E}(D) \).