A nonzero left-null datum passes the series test but is not in the range

The standard dot products on $\mathbb{R}$ and ${\mathbb{R}}^2$ satisfy the inner-product axioms [F3] by coordinatewise addition, multiplication and ${\sum }_i{x}_i^2=0\Rightarrow x_i=0$ for each coordinate. The standard Euclidean spaces $\mathbb{R}$ and ${\mathbb{R}}^2$ are complete: if a sequence is Cauchy in Euclidean norm, each coordinate is Cauchy because its absolute difference is bounded by the Euclidean distance; each coordinate converges by [F4], and the finite sum of squared coordinate errors tends to zero. Thus they are real Hilbert spaces by [F3] and [F5]. The map $K$ is linear and $\mathrm{\parallel }Kx\mathrm{\parallel }=\mathrm{\mid }x\mathrm{\mid }$, so it is bounded; its range is the one-dimensional span of $(1,0)$, and [F2] makes it compact. It therefore gives the stated compact linear inverse model by [F1].

For $a\in \mathbb{R}$ and $(b,c)\in {\mathbb{R}}^2$, $⟨Ka,(b,c)⟩=ab=⟨a,K^{\ast }(b,c)⟩$. Hence $N(K^{\ast })=\{(0,c):c\in \mathbb{R}\}$, while $R(K)=\{(a,0):a\in \mathbb{R}\}$. In particular $y$ is a nonzero element of $N(K^{\ast })$, and no scalar $x$ satisfies $Kx=y$.

The vector $v_1=1$ spans the input and $u_1=(1,0)$ spans $\overline{R(K)}$; they satisfy $K{v}_1=u_1$ and $K^{\ast }{u}_1=v_1$. Thus the positive singular pair has $s_1=1$. The Picard sum defined in the statement has sole term $\mathrm{\mid }⟨(0,1),(1,0)⟩{\mathrm{\mid }}^2\mathrm{/}{1}^2=0$. Step 1.2 shows that $y\in N(K^{\ast })\setminus \{0\}$, so the missing orthogonality condition allows finite Picard sum without exact solvability.