= Discontinuous functional detected by vanishing sequence tails
{title2=$f(v_N)=1,\quad\|v_N\|_2\to0$}
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 $w=(2^{-n})_{n\geq1}$. Define $f(w)=1$ and $f(e_n)=0$. The tails $v_N=w-\sum_{n=1}^N2^{-n}e_n$ have norm $2^{-N}/\sqrt3$ but image one. Thus the functional vanishes on the dense finite-support subspace while being nonzero elsewhere.
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