I'm reading Sheldon Axler's Measure theory book, and in a subsection dedicated to the Hahn-Banach Theorem, he states:
In this subsection, we show that infinite-dimensional normed vectorspaces have plenty of continuous linear functional. We do did byshowing that a bounded linear functional on a subspace of a normedvector space can be extended to a bounded linear functional on thewhole space without increasing its norm.
I don't understand how the Hahn-Banach theorem can be used to show that every infinite-dimensional normed vector have 'plenty' (how does one define this?) of continuous/bounded linear functionals.