Let $X$ be a set of cardinality $\kappa$ such that $\kappa^\omega=\kappa$. We
prove that the linear algebra $\mathbb{R}^X$ (or $\mathbb{C}^X$) contains a
free linear algebra with $2^\kappa$ generators. Using this, we prove several
algebrability results for spaces $\mathbb{C}^\mathbb{C}$ and
$\mathbb{R}^\mathbb{R}$. In particular, we show that the set of all perfectly
everywhere surjective functions $f:\mathbb{C}\to\mathbb{C}$ is strongly
$2^\mathfrak{c}$-algebrable. We also show that the set of all functions
$f:\mathbb{R}\to\mathbb{R}$ whose sets of continuity points equals some fixed
$G_\delta$ set $G$ is strongly $2^\mathfrak{c}$-algebrable if and only if
$\mathbb{R}\setminus G$ is $\mathfrak{c}$-dense in itself
Any operator in a von Neumann algebra is a linear combination of a finite number of projections from the algebra with coefficients from the center of the algebra. Those von Neumann algebras that are the complex linear span of their projections are identified.
We characterize sequences of numbers (a n ) such that n≥1 a n Φ n converges a.e. for any orthonormal system (Φ n ) in any L 2 -space. In our criterion, we use the set B = { m≥n |a m | 2 ; n ≥ 1} and its information function
Let H be an infinite dimensional Hilbert space. We show that there exist three orthogonal projections X 1 , X 2 , X 3 onto closed subspaces of H such that for every 0 = z 0 ∈ H there exist k 1 , k 2 , · · · ∈ {1, 2, 3} so that the sequence of iterates defined by z n = X kn z n−1 does not converge in norm.
Any operator in a von Neumann algebra is a linear combination of a finite number of projections from the algebra with coefficients from the center of the algebra. Those von Neumann algebras that are the complex linear span of their projections are identified.
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