This chapter introduces the fundamental concepts of topological vector spaces (t.v.s.). The scalar field K is an arbitrary, non-discrete valued field, with the uniformity derived from its absolute value. The purpose is to highlight the essential properties of real and complex fields. Section 1 discusses vector space topologies in terms of neighborhoods of 0 and the associated uniformity. Section 2 covers methods for constructing new t.v.s. from existing ones. Section 3 collects standard tools for finite-dimensional spaces, followed by a brief discussion of affine subspaces and hyperplanes. Section 5 focuses on the concept of boundedness, while Section 6 addresses metrizability, which is important for applications in analysis and the history of the subject. Section 7 discusses the transition between real and complex fields. A topological vector space is a vector space with a topology that makes addition and scalar multiplication continuous. The pair (L, T) is a t.v.s. over K if addition and scalar multiplication are continuous. A t.v.s. is a commutative topological group. Isomorphic t.v.s. are those that have a bijective linear homeomorphism between them. Key properties of t.v.s. include: (i) translations by scalar multiples are homeomorphisms; (ii) the closure of a set is the intersection of translates of neighborhoods of 0; (iii) the sum of an open set and any set is open; (iv) the sum of two closed sets, one of which is compact, is closed; (v) the closure and interior of a circled set are circled. These properties follow from the continuity of addition and scalar multiplication.This chapter introduces the fundamental concepts of topological vector spaces (t.v.s.). The scalar field K is an arbitrary, non-discrete valued field, with the uniformity derived from its absolute value. The purpose is to highlight the essential properties of real and complex fields. Section 1 discusses vector space topologies in terms of neighborhoods of 0 and the associated uniformity. Section 2 covers methods for constructing new t.v.s. from existing ones. Section 3 collects standard tools for finite-dimensional spaces, followed by a brief discussion of affine subspaces and hyperplanes. Section 5 focuses on the concept of boundedness, while Section 6 addresses metrizability, which is important for applications in analysis and the history of the subject. Section 7 discusses the transition between real and complex fields. A topological vector space is a vector space with a topology that makes addition and scalar multiplication continuous. The pair (L, T) is a t.v.s. over K if addition and scalar multiplication are continuous. A t.v.s. is a commutative topological group. Isomorphic t.v.s. are those that have a bijective linear homeomorphism between them. Key properties of t.v.s. include: (i) translations by scalar multiples are homeomorphisms; (ii) the closure of a set is the intersection of translates of neighborhoods of 0; (iii) the sum of an open set and any set is open; (iv) the sum of two closed sets, one of which is compact, is closed; (v) the closure and interior of a circled set are circled. These properties follow from the continuity of addition and scalar multiplication.