Linear Functional on a Linear Subspace

In the setting of Separable Real Normed Spaces and Hahn–Banach: Notation, let $V$ be a real vector space, let $W$ be a linear subspace of $V$, and let $f:W\to\mathbb{R}$ be a function. The sums and multiples below lie in $W$ by Linear Subspace of a Real Vector Space (ii) and Linear Subspace of a Real Vector Space (iii).

(i) Additivity. $f(u+v)=f(u)+f(v)$ for all $u,v\in W$.

(ii) Homogeneity. $f(\lambda u)=\lambda f(u)$ for all $\lambda\in\mathbb{R}$ and $u\in W$.

(iii) Linear functional. $f$ is a linear functional on $W$ if it satisfies (i) and (ii). A linear functional on $V$ is one on the linear subspace $W=V$ ($V$ is a linear subspace of itself, Linear Subspace of a Real Vector Space (i), Linear Subspace of a Real Vector Space (ii) and Linear Subspace of a Real Vector Space (iii) holding because $0_{V}\in V$ and the addition and scalar multiplication of $V$ take values in $V$).