This paper explores the minimum-time optimal control problem in a locally convex linear topological space 𝐿. To solve this problem, we assume that 𝐿 is Hausdorff and that the continuous time system is controllable. The system is represented by Σ = {𝐼, 𝐿, 𝐸_π, 𝑋, 𝑌, ϕ, ψ}, where 𝐼 = [𝑡₀, 𝑡], 𝑋 is the state space of state variables and 𝑌 is the output space. In this system Σ, we take the unit ball in 𝐿 as the polar set 𝐸_π, which characterises the locally convex linear topological space 𝐿. By employing the concept of this unit ball and defining the continuous linear operator 𝑇_𝑡: 𝐸_π → 𝑋 for each fixed time 𝑡 ∈ 𝐼, we first establish the existence of an optimal control 𝑙 ∈ 𝐸_π by proving the compactness of the attainable set 𝐴(𝑡). Next, we obtain the minimum-time optimal control by considering a decreasing sequence {𝑡_n} → 𝑡 = 𝑡(1) and by considering a sequence {𝑙ₙ} in 𝐸_π such that {𝑙ₙ} → 𝑙. Thus, the concept of this unit ball is a novel technique in our study. Furthermore, we approach Pontryagin's maximum principle for the existence of the minimum-time optimal control 𝑙 for fixed 𝑡 ∈ 𝐼 minimizing the time cost function 𝐽_𝑡 by maximizing the Hamiltonian 𝐻, considering the differential equation 𝑢̇ = 𝐴𝑢(𝑡) + 𝐵𝑙(𝑡) represented by the dynamical system Σ. Additionally, we provide a geometric interpretation for maximizing the Hamiltonian 𝐻 to obtain the minimum-time optimal control 𝑙_opt at time 𝑡 = 𝑡_opt = 𝑡(1). Finally, we provide two examples based on Pontryagin's maximum principle related to our problem.