Abstract
A measurable function f defined on a measurable subset A of the real line R is called pth power summable on A if │f│$^{p}$ is integrable on A and the set of all pth power summable functions on A is denoted by L$^{p}$ (A). For each member f in L$^{p}$ (A), we define ∥f∥$_{p}$ =(equation omitted) For real numbers p and q where (equation omitted) and (equation omitted), we discuss the Holder's inequality ∥fg∥$_1$<∥f∥$_{p}$ ∥g∥$_{q}$ , f$\in$L$^{p}$ (A), g$\in$L$^{q}$ (A) and the Minkowski inequality ∥＋g∥$_{p}$ <∥f∥$_{p}$ ＋∥g∥$_{p}$ , f,g$\in$L$^{p}$ (A). In this paper also discuss that L$_{p}$ (A) becomes a metric space with the metric $\rho$ : L$^{p}$ (A) $\times$L$^{p}$ (A) longrightarrow R where $\rho$(f,g)=∥f-g∥$_{p}$ , f,g$\in$L$^{p}$ (A). Then, in this paper prove the Riesz-Fischer theorem, i.e., the space L$^{p}$ (A) is complete and that the space L$^{p}$ (A) is separable.