LpmathbbRn

LpmathbbRn refers to the space of Lebesgue integrable functions on the n-dimensional Euclidean space, denoted as Rn. This space is a fundamental concept in the field of functional analysis and is widely used in various areas of mathematics and physics. The space LpmathbbRn consists of all measurable functions f: Rn → R such that the Lebesgue integral of the p-th power of the absolute value of f is finite. This is expressed mathematically as:

∫|f(x)|^p dx < ∞

where dx denotes the Lebesgue measure on Rn. The parameter p is a positive real number, and the space is often denoted as Lp(Rn) or simply Lp when the context is clear. The space LpmathbbRn is equipped with a norm, which is defined as:

||f||p = (∫|f(x)|^p dx)^(1/p)

for 1 ≤ p < ∞, and

||f||∞ = ess sup |f(x)|

for p = ∞. The space LpmathbbRn is a Banach space for 1 ≤ p ≤ ∞, meaning it is complete with respect to the norm defined above. This space plays a crucial role in the study of partial differential equations, harmonic analysis, and probability theory.