According to (Nelson 2008)1 P58

  • base
    1. (of a number system) The number represented by the numeral ‘10’ in a positional number system. Thus, in the decimal system the base (ten) is represented by 10; in a binary system the base (two) is also represented by 10.
    2. (of logarithms) The number which, raised to the power of a given logarithm, produces a given number. Thus, if the logarithm of x to base b (written as logb x) is y, then by = x.
    3. A line or plane in a geometric figure relative to which the altitude of the figure is measured.
  • basis pl. bases: A subset of a vector space that is linearly independent and spans the space.

Therefore, when we talk about the basis of some space, such as an Euclidean space or Sobolev space, the word basis itself is already a collection of elements in this space, while any single one of them should be described as basis element or basis function. On the other hand, base should never appear in such occasions. Also note the plural form of basis coincides with that of base.

  1. Nelson, David, ed. 2008. The Penguin Dictionary of Mathematics: Fourth Edition. 4th edition. London: Penguin Books. ↩