This paper presents fixed-point and minimax theorems in locally convex topological linear spaces. Ky Fan generalizes Kakutani's and Tychonoff's fixed-point theorems to locally convex spaces. The main result, Theorem 1, states that for any upper semicontinuous point-to-set transformation from a compact convex set in a locally convex space into the family of closed convex subsets of the set, there exists a fixed point. The proof involves constructing a sequence of sets and showing their non-emptiness and closure. Theorem 2 extends this result to families of compact convex sets and shows that the intersection of closed subsets is non-empty under certain conditions. Theorem 3 extends von Neumann's minimax theorem to locally convex spaces, showing that the maximum of the minimum equals the minimum of the maximum for a continuous function on the product of two compact convex sets. The paper also discusses applications of these theorems, including a result on systems of linear equations with dominant diagonal coefficients. The theorems are supported by references to previous works by Kakutani, Tychonoff, von Neumann, and others. The paper emphasizes the importance of upper semicontinuity and the use of uniform spaces in the proofs.This paper presents fixed-point and minimax theorems in locally convex topological linear spaces. Ky Fan generalizes Kakutani's and Tychonoff's fixed-point theorems to locally convex spaces. The main result, Theorem 1, states that for any upper semicontinuous point-to-set transformation from a compact convex set in a locally convex space into the family of closed convex subsets of the set, there exists a fixed point. The proof involves constructing a sequence of sets and showing their non-emptiness and closure. Theorem 2 extends this result to families of compact convex sets and shows that the intersection of closed subsets is non-empty under certain conditions. Theorem 3 extends von Neumann's minimax theorem to locally convex spaces, showing that the maximum of the minimum equals the minimum of the maximum for a continuous function on the product of two compact convex sets. The paper also discusses applications of these theorems, including a result on systems of linear equations with dominant diagonal coefficients. The theorems are supported by references to previous works by Kakutani, Tychonoff, von Neumann, and others. The paper emphasizes the importance of upper semicontinuity and the use of uniform spaces in the proofs.