Corollary 3

— Let $V$ be a vector subspace of the normed space $E$ and let $f$ be a continuous linear form over $V$; then there exists a continuous linear form $h$ defined over $E$ which extends $f$ and is of the same norm (GT, X, § 3.2).

Apply cor. 1, taking $p(x) = \|f\| \cdot \|x\|$, which gives $\|h\| \leq \|f\|$; but clearly $\|h\| \geq \|f\|$, and the corollary follows.

The conclusion of cor. 3 is not necessarily valid for continuous linear mappings of a normed space into an arbitrary normed space (IV, p. 55, exerc. 16, c) and V, p. 65, exerc. 22).