All continuous linear functionals in a linear normed space form a complete linear normed space according to the norm ‖f‖=sup|f(x)|(‖x‖= 1), which is the * * * yoke space of the original linear normed space. The construction of dual space is the abstraction of the relationship between row vector (1×n) and column vector (n× 1).

So the difference between * * * yoke space and dual space is that all continuous linear functional groups in a linear normed space form a complete linear normed space according to the norm ‖f‖=sup|f(x)|(‖x‖= 1), which is * * of the original linear normed space. The construction of dual space is the abstraction of the relationship between row vector (1×n) and column vector (n× 1).

dual space

In mathematics, any vector space V has its corresponding dual vector space (or dual space for short), which is composed of the linear functional of V. This dual space has the structure of general vector space, such as vector addition and scalar multiplication. The dual space defined in this way can also be called algebraic dual space. In the case of topological vector space, the dual space composed of continuous linear functional is called continuous dual space.

Dual space is an abstraction of the relationship between row vector (1×n) and column vector (n× 1). This structure can be carried out in infinite dimensional space, and provides an important point of view for measurement, distribution and Hilbert space. The application of dual space is the characteristic of functional analysis theory. Fourier transform also contains the concept of dual space.

Definition (linear function)

Let e be a normed space and L(E, r) be a real Banach space composed of continuous linear mappings from e to r, which is called the dual space of e.