The paragraph is given in the following pictures: If $A$ is an operator from $H_{1}$ to $H_{2}$, The paragraph is divided into 5 sentences. I could not understand the first, second, forth and fifth sentences. Could anyone explain them for me please? Thanks!
Let $\mathfrak{F}(S)$ denotes the set of filters (including the improper filter) on a poset $S$, ordered reversely to set theoretic inclusion of filters. Let $Da$ for a lattice element $a$ denote its sublattice $\{ x \mid x \leq a \}$. Let $Z(X)$ denotes the set of complemented elements of the la...
Let $T:l_{2} \rightarrow l_{2}$ be defined by $$T(x) = \left(x_{1},\frac{x_{2}}{2},\frac{x_{3}}{4},\ldots,\frac{x_{n}}{2^{n-1}},\ldots \right)$$ where $x = (x_{1},x_{2},.....).$ Find Im$T$. I know that the kernel of this operator is $\{0\}$ and I know that the sum of the dim Im$T$ + dim ker$...
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