Singular values equal approximation numbers

Given: Countable Choice, a compact $T:H\to K$, its singular system $(e_j{)}_{j\in J}$, $(f_j{)}_{j\in J}$, $(s_j{)}_{j\in J}$ from the singular-value decomposition, and the numbers $a_n(T)$.

Singular-value decomposition. With the index set $J$ of the positive singular values with multiplicity, there are orthonormal systems $(e_j{)}_{j\in J}\subseteq (\ker T{)}^{\perp }$ and $(f_j{)}_{j\in J}\subseteq \overline{\mathrm{ran}T}$ with $\mathrm{\mid }T\mathrm{\mid }{e}_j=s_j{e}_j$ and $f_j=s_j^{-1}T{e}_j$, the expansion $Tx={\sum }_{j\in J}{s}_j⟨x,e_j⟩f_j$ holds in norm, and for every $n$ with $n+1\in J$ the partial sum $T_n={\sum }_{j\le n}{s}_j⟨\cdot ,e_j⟩f_j$ satisfies $\mathrm{\parallel }T-T_n\mathrm{\parallel }\le s_{n+1}$; moreover $s_n(T)=0$ for all $n>r$ when $r={\dim }_{\mathbb{F}}\mathrm{ran}T<+\mathrm{\infty }$, and $J=\{1,\dots ,r\}$ in that case (Singular value decomposition for compact operators, Absolute value and singular values of a compact operator).

Ranks of truncations. A finite sum ${\sum }_{j\in F}{s}_j⟨\cdot ,e_j⟩f_j$ over a finite $F$ has range contained in the span of the finitely many $f_j$, hence rank at most $\mathrm{\mid }F\mathrm{\mid }$; for $F=\{1,\dots ,n-1\}$ the truncation $T_{n-1}$ therefore has rank $<n$ (Finite-dimensional vector space, and its dimension ${\dim }_{F}V$; infinite-dimensional means having no finite basis, Kernel and image of a linear map).

Rank–nullity in finite dimensions. A linear map of a finite-dimensional space $V$ satisfies $\dim V=\dim \ker F+\dim \mathrm{ran}F$; hence a linear map on a finite-dimensional space of dimension $n$ with rank $<n$ has a nonzero kernel (Rank-nullity: ${\dim }_{F}V=\mathrm{nullity}T+\mathrm{rank}T$, Finite-dimensional vector space, and its dimension ${\dim }_{F}V$; infinite-dimensional means having no finite basis, Kernel and image of a linear map).

Orthonormal expansion in the span. If $x={\sum }_{j\in F}{c}_j{e}_j$ lies in the span of finitely many members of the orthonormal family $(e_j)$, then $\mathrm{\parallel }x{\mathrm{\parallel }}^2={\sum }_{j\in F}\mathrm{\mid }{c}_j{\mathrm{\mid }}^2$, $⟨x,e_j⟩=c_j$ for $j\in F$ and $⟨x,e_j⟩=0$ for $j\mathrm{\notin }F$; in particular, for $x$ in the span of $e_1,\dots ,e_n$ the expansion of $Tx$ reduces to the finite sum ${\sum }_{j\le n}{s}_j⟨x,e_j⟩f_j$; the finite Bessel inequality bounds partial sums of coefficients (Orthonormal families, complete orthonormal systems and Hilbert bases, The finite Bessel inequality and best approximation by a finite orthonormal family, Real and complex inner-product spaces and their induced length).

Infimum. Every nonempty lower-bounded subset of $\mathbb{R}$ has an infimum (Every nonempty set bounded below has an infimum); by the defining greatest-lower-bound property, every lower bound $a$ of $S$ satisfies $a\le \inf S$, and conversely $a\le \inf S$ makes $a$ a lower bound of $S$ (Greatest lower bound (infimum)).

Countable Choice is the standing hypothesis of this pair's Hilbert-space interface (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)).