<P> Along the same lines, one can prove that x and y are distinct by finding some function f and proving that f (x) and f (y) are distinct . This may seem like a simple idea, and it is, but many deep results in mathematics concern when you can prove distinctness by particular methods . For example, </P> <Ul> <Li> The Hahn--Banach theorem says (among other things) that distinct elements of a Banach space can be proved to be distinct using only linear functionals . </Li> <Li> In category theory, if f is a functor between categories C and D, then f always maps isomorphic objects to isomorphic objects . Thus, one way to show two objects of C are distinct (up to isomorphism) is to show that their images under f are distinct (i.e. not isomorphic). </Li> </Ul> <Li> The Hahn--Banach theorem says (among other things) that distinct elements of a Banach space can be proved to be distinct using only linear functionals . </Li> <Li> In category theory, if f is a functor between categories C and D, then f always maps isomorphic objects to isomorphic objects . Thus, one way to show two objects of C are distinct (up to isomorphism) is to show that their images under f are distinct (i.e. not isomorphic). </Li>

What is the mean of distinct in hindi