Real Hilbert Space

Let $E$ be a real inner product space with distance $d$, which is a metric on $E$ by The Norm Metric of a Real Inner Product Space: Triangle Inequalities, Limits and Continuity §metric.

1. (Hilbert space)¶ $E$ is a real Hilbert space if the metric space $(E,d)$ is complete.

2. (Topological vocabulary)¶ Whenever a subset of a real inner product space $E$ is called open, closed, dense or bounded, or a sequence in $E$ is called convergent or Cauchy, this refers to the metric space $(E,d)$: a subset is open if it is open in $(E,d)$, and these open sets form a topology $\mathcal{T}_{d}$ on $E$ by Metric Open Sets Form a Topology; a subset is closed if it is closed in $(E,\mathcal{T}_{d})$, its closure is its closure in $(E,\mathcal{T}_{d})$, and it is dense if it is dense in $(E,\mathcal{T}_{d})$; a subset is bounded if it is bounded in $(E,d)$; and a sequence in $E$ is convergent or Cauchy if it is convergent or Cauchy in $(E,d)$.

3. (Closed linear subspace)¶ A closed linear subspace of $E$ is a linear subspace of $E$ that is closed in the sense of clause 2.