$p$-Summable Sequence Spaces with Inner Products

Abstract

We revisit the space $\ell^p$ of $p$-summable sequences of real
numbers. In particular, we show that this space is actually
contained in a (weighted) inner product space. The relationship
between $\ell^p$ and the (weighted) inner product space that
contains $\ell^p$ is studied. For $p>2$, we also obtain a result
which describe how the weighted inner product space is associated
to the weights.

Keywords

• inner product spaces, normed spaces, p -summable sequences, weights

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