This paper introduces the Hilbert space-valued Wiener process and the corresponding Itô stochastic integral. It uses semigroup theory to establish the existence and uniqueness of weak solutions for linear and semilinear stochastic evolution equations in Hilbert space. The abstract theory is then applied to the linear heat and wave equations driven by additive noise. The paper covers functional analysis essentials, elements of Banach space-valued stochastic analysis, stochastic integrals for nuclear and cylindrical Wiener processes, and stochastic evolution equations with additive noise. It includes examples of the heat and wave equations. The paper also discusses pseudo-inverse operators, the Cameron-Martin space, and properties of Gaussian measures. The main results include the characterization of Gaussian measures, the representation of Gaussian random variables, and the construction of Q-Wiener processes. The paper concludes with the existence of Gaussian measures and the representation of Q-Wiener processes in terms of independent Brownian motions.This paper introduces the Hilbert space-valued Wiener process and the corresponding Itô stochastic integral. It uses semigroup theory to establish the existence and uniqueness of weak solutions for linear and semilinear stochastic evolution equations in Hilbert space. The abstract theory is then applied to the linear heat and wave equations driven by additive noise. The paper covers functional analysis essentials, elements of Banach space-valued stochastic analysis, stochastic integrals for nuclear and cylindrical Wiener processes, and stochastic evolution equations with additive noise. It includes examples of the heat and wave equations. The paper also discusses pseudo-inverse operators, the Cameron-Martin space, and properties of Gaussian measures. The main results include the characterization of Gaussian measures, the representation of Gaussian random variables, and the construction of Q-Wiener processes. The paper concludes with the existence of Gaussian measures and the representation of Q-Wiener processes in terms of independent Brownian motions.