Regular matrix summability method

A matrix summability method is regular if it preserves the ordinary limit of every convergent sequence. The Silverman-Toeplitz theorem characterizes regularity by uniformly bounded absolute row sums, entrywise convergence to zero down every fixed column, and convergence of the row sums to one.

Selecting a subsequence is a regular matrix summability method: every row has one entry equal to one, every fixed column eventually vanishes, and all row sums equal one. Every bounded sequence of real or complex numbers has a convergent subsequence by the Bolzano-Weierstrass theorem, so each bounded sequence is summable by some sequence-dependent regular method. The order of the quantifiers matters: no single regular matrix sums all bounded sequences.

The summability domain is the inverse image of the convergent sequence space under the bounded matrix operator on the l-infinity sequence space. It is therefore a closed linear subspace. The bounded divergence for every regular summability matrix makes it proper, so it has empty interior and is a nowhere dense set. The Baire category theorem then ensures that countably many regular methods miss some common bounded sequence.

For every regular summability matrix $A$, there is a bounded sequence whose transformed row sums do not converge. A gliding hump argument selects successive rows whose mass on already assigned coordinates is small and whose remaining tails are small. On each new finite block, choose signs agreeing or disagreeing with that row alternately. Since row sums tend to one, the transformed values have subsequences uniformly above and below zero. This does not contradict preservation of ordinary convergence, since the constructed sequence is divergent.

For a real or complex matrix, the three displayed conditions are equivalent to being a regular matrix summability method. For necessity, first apply the Uniform boundedness principle to each row's finite partial sums on the convergent sequence space, obtaining absolute summability of that row. Apply the principle again to the full row functionals, obtaining a uniform bound. Coordinate vectors and the constant-one sequence give the remaining conditions. For sufficiency, write $x_j=L+(x_j-L)$ and split the error into a finite head, controlled by columnwise convergence, and a uniformly small tail, controlled by the absolute row bound.