A finite dimensional real l^p space is a mathematical concept that describes a vector space of functions or sequences with finite dimensions and with elements that are real numbers. It is often used in the study of functional analysis and measure theory.
The main difference between l^p space and finite dimensional real l^p space is that l^p space can have infinite dimensions, while finite dimensional real l^p space has a fixed, finite number of dimensions. In other words, l^p space allows for infinite linear combinations of elements, while finite dimensional real l^p space only allows for a finite number of linear combinations.
Some examples of finite dimensional real l^p spaces include Euclidean spaces (such as R^n), function spaces (such as C([0,1])), and sequence spaces (such as l^p([1,n])). These spaces can have different dimensions and different values of p, depending on the specific problem or application.
The norm in finite dimensional real l^p spaces is defined as ||x||_p = (|x_1|^p + |x_2|^p + ... + |x_n|^p)^(1/p), where x = (x_1, x_2, ..., x_n) is a vector in the space. This norm measures the "size" or "length" of a vector in the l^p space and is used to define concepts such as convergence and continuity.
Finite dimensional real l^p spaces have many applications in mathematics, physics, and engineering. They are used to study optimization problems, control theory, signal processing, and numerical analysis, among others. They are also used as a foundation for more advanced concepts such as Banach spaces and Hilbert spaces.