Angle in the space of $p$-summable sequences

The aim of this paper is to investigate completness of $A$ that equipped with usual norm on $p$-summable sequences space where $A$ is subspace in $p$-summable sequences space and $1\le p<\mathrm{\infty }$. We also introduce a new inner product on $A$ and prove completness of $A$ using a new norm that corresponds this new inner product. Moreover, we discuss the angle between two vectors and two subspaces in $A$. In particular, we discuss the angle between $1$-dimensional subspace and $(s-1)$-dimensional subspace where $s\ge 2$ of $A$.