basis,
n. 1. in a vector space, any maximal independent set or minimal spanning set.
b. any linearly independent subset of a vector space that spans the space. The cardinality of such a subset is the di mension of the space. For example, the dimension of the vector space of all polynomials over a field is ℵ0, and the set with elements 1, x, x2,..., xn,..., forms a basis; the vectors (1,0,0), (0,1,0), (0,0,1) are a basis for three-dimensional Euclidean space.
c. (in a free module) any linearly independent set that spans the module. 2. a Schauder basis of a separable normed linear space is a sequence of vectors {xi} in terms of which each element of the space can be expressed uniquely as a countab le norm-convergent sum:
.
See Schauder basis problem.
3. a Hamel basis of a vector space is a linearly independent set whose finite linear combinations span the space.
4. an orthonormal basis of an inner product space is a basis comprised of orthonormal vectors.
b. any linearly independent subset of a vector space that spans the space. The cardinality of such a subset is the di mension of the space. For example, the dimension of the vector space of all polynomials over a field is ℵ0, and the set with elements 1, x, x2,..., xn,..., forms a basis; the vectors (1,0,0), (0,1,0), (0,0,1) are a basis for three-dimensional Euclidean space.
c. (in a free module) any linearly independent set that spans the module. 2. a Schauder basis of a separable normed linear space is a sequence of vectors {xi} in terms of which each element of the space can be expressed uniquely as a countab le norm-convergent sum:
.