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Register a new userThe Infinite Gauss-Jordan Elimination on Row-Finite omega x omega Matrices
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Authors
Alexandros G. Paraskevopoulos
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The Gauss-Jordan elimination algorithm is extended to reduce a row-finite $omegatimesomega$ matrix to lower row-reduced form, founded on a strategy of rightmost pivot elements. Such reduced matrix form preserves row equivalence, unlike the dominant (upper) row-reduced form. This algorithm provides a constructive alternative to an earlier existence and uniqueness result for Quasi-Hermite forms based on the axiom of countable choice. As a consequence, the general solution of an infinite system of linear equations with a row-finite coefficient $omegatimesomega$ matrix is fully constructible.
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The construction of the general solution sequence of row-finite linear systems is accomplished by implementing -ad infinitum- the Gauss-Jordan algorithm under a rightmost pivot elimination strategy. The algorithm generates a basis (finite or Schauder) of the homogeneous solution space for row-finite systems. The infinite Gaussian elimination part of the algorithm solves linear difference equations with variable coefficients of regular order, including equations of constant order and of ascending order. The general solution thus obtained can be expressed as a single Hessenbergian.
A locally convex space (lcs) $E$ is said to have an $omega^{omega}$-base if $E$ has a neighborhood base ${U_{alpha}:alphainomega^omega}$ at zero such that $U_{beta}subseteq U_{alpha}$ for all $alphaleqbeta$. The class of lcs with an $omega^{omega}$-base is large, among others contains all $(LM)$-spaces (hence $(LF)$-spaces), strong duals of distinguished Frechet lcs (hence spaces of distributions $D(Omega)$). A remarkable result of Cascales-Orihuela states that every compact set in a lcs with an $omega^{omega}$-base is metrizable. Our main result shows that every uncountable-dimensional lcs with an $omega^{omega}$-base contains an infinite-dimensional metrizable compact subset. On the other hand, the countable-dimensional space $varphi$ endowed with the finest locally convex topology has an $omega^omega$-base but contains no infinite-dimensional compact subsets. It turns out that $varphi$ is a unique infinite-dimensional locally convex space which is a $k_{mathbb{R}}$-space containing no infinite-dimensional compact subsets. Applications to spaces $C_{p}(X)$ are provided.
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