Remark

— If $g$ is a continuous linear mapping of M in the product space $\mathbf{R}^I$, then there exists a continuous linear mapping $h$ of E in $\mathbf{R}^I$ that is an extension of $g$; for writing $g = (g_i)$, where the $g_i$ are continuous linear forms defined over M, there is an extension $h_i$ of $g_i$ for each $i \in I$, such that $h_i$ is a continuous linear form over E. The continuous linear mapping $h = (h_i)$ has the required properties.

Note that if F is a locally convex Hausdorff space and $g$ a continuous linear mapping of M in F, then there does not necessarily exist a continuous linear mapping of E in F which is an extension of $g$ (IV, p. 55, exerc. 16, c)). However there does exist such an extension when M is finite dimensional (cf. cor. 2, below).