A linear functional is a specific type of function that maps vectors from a vector space to the underlying field (usually the real or complex numbers) while satisfying linearity properties. This means that for any vectors \( x \) and \( y \) in the vector space and any scalar \( c \), a linear functional \( f \) satisfies \( f(x + y) = f(x) + f(y) \) and \( f(cx) = c f(x) \). Linear functionals are central to understanding dual spaces and play a significant role in various mathematical contexts, particularly in the Riesz representation theorem, which connects linear functionals to inner products on Hilbert spaces.
congrats on reading the definition of linear functional. now let's actually learn it.