Let X be a normed space, let Y⊂X be a subspace. Then any continuous linear functional
λ∈Y∗ on Y can be extended to a continuous linear functional λ^∈X∗
on X with the same operator norm, thus λ^ agrees with λ on Y
and ∣∣λ^∣∣X∗=∣∣λ∣∣Y∗.
Note that λ^ is not necessarily unique.