Abstrakt
We simplify the presentation of the method of elementary submodels and we show that it can be used for simplifying proofs of existing separable reduction theorems and for obtaining new ones. Given a nonseparable Banach space $X$ and either a subset $A\subset X$ or a function $f$ defined on $X$, we are able for certain properties produce a separable subspace of $X$ which determines whether $A$ or $f$ has the property.
Such results are proved for properties of sets ''to be dense, nowhere dense, meager, residual or porous'' and for properties of functions ''to be continuous, semicontinuous or Fr\'echet differentiable''. Our method of creating separable subspaces enables us to combine our results, so we easily get separable reductions of properties such as ''to be continuous on a dense subset'', ''to be Fr\'echet differentiable on a residual subset'', etc.
Finally, we show some applications of presented separable reduction theorems and demonstrate that some results of Zaj\'{\i}\v{c}ek, Lindenstrauss and Preiss hold in nonseparable setting as well.