Linear functional

A linear functional is a linear map from a vector space to its scalar field.

For a nonzero Euclidean vector $w$, the Cauchy-Schwarz inequality gives $z\cdot w\ge -\Vert w\Vert$ when $\Vert z\Vert =1$. Equality is attained uniquely at $z=-w/\Vert w\Vert$. On a sphere of radius $r>0$ the minimizer is $-rw/\Vert w\Vert$, and the minimum is $-r\Vert w\Vert$. This turns many constrained vector problems into a normalization of one fixed vector. If $w=0$, every point minimizes the functional.

If a linear functional on a normed vector space has $f(v_N)=1$ for a sequence tending to zero in norm, it is discontinuous and unbounded. An explicit example is the algebraic span in ${\ell }^2$ of the coordinate vectors and $w=(2^{-n}{)}_{n\ge 1}$. Define $f(w)=1$ and $f(e_n)=0$. The tails $v_N=w-{\sum }_{n=1}^N{2}^{-n}{e}_n$ have norm $2^{-N}/\sqrt{3}$ but image one. Thus the functional vanishes on the dense finite-support subspace while being nonzero elsewhere.