medium
2 min read
Double dual
Double Dual Spaces and Annihilators The double dual of a linear space $V$ is another linear space $V^ $, equipped with the dual pairing between vectors a...
Quick Actions
Insights
Difficultymedium
Reading Time2 min
Double Dual Spaces and Annihilators The double dual of a linear space $V$ is another linear space $V^ $, equipped with the dual pairing between vectors a...
The double dual of a linear space is another linear space , equipped with the dual pairing between vectors and linear transformations. This means the following:
A linear transformation is represented as a linear map .
The inner product on becomes the dual pairing between vectors: .
The norm on becomes the norm of the vector: ||x|| = sqrt((x,x)) for all vectors x in V.
The dual space inherits the norm structure from , making it a normed linear space.
Examples:
Consider the vector space of polynomials of degree 2 with real coefficients, V = span{1, x, x^2}.
The dual space V* is the space of polynomials of degree less than 2, V^* = span{x, x^2, x^3}.
The linear transformation T(x) = x^2 is represented as a linear map in V by T(x) = x^2.
The dual pairing between x and T(x) expresses the linear transformation using the inner product.
The norm on V can be chosen to be ||x|| = sqrt(x_1^2 + x_2^2 + ... + x_n^2) for all vectors x in V. This norm leads to the standard Euclidean norm on the vector space.
Similarly, the norm on V* can be chosen to be ||T(x)|| = sqrt(T(x)_1^2 + T(x)_2^2 + ... + T(x)_n^2) for all vectors x in V*.
The double dual is a powerful tool in linear algebra, as it allows us to manipulate linear transformations and analyze their properties. By studying the dual space, we can gain insights into the behavior of the original space and its linear transformations