1. V is a vector space. The zero vector is the function z(x) = 0.

  2. F is a vector space.

  3. W is not a vector space. Neither of the two proposed operations of vector addition and scalar multiplication works.

    For example, if f(1) = 2 and g(1) = 2 then (f + g)(1) = 4 and so f + g is not in W.

    There is no zero vector in W either.

Copyright c 1995 by Frank Wattenberg, Department of Mathematics, Montana State University, Bozeman, MT 59717