# Linear subspace - closed subspace - new inner product

"Let $$(H, \rho)$$ be a Hilbert space, and let $$\mathscr{S} \subseteq H$$ be a linear subspace. Let $$\mathscr{P}_{\mathscr{S}} s$$ denote the orthogonal projection operator onto the closed linear subspace $$\overline{\mathscr{S}}^{\rho}$$ in $$(H, \rho) .$$ We also suppose that $$\tau$$ is an inner product defined over $$\mathscr{S}$$ such that $$(\mathscr{S}, \tau)$$ has structure of Hilbert space."

From this statement I would like to clarify the following:

1. A linear subspace is not necessarily finite dimensional space, it can be infitite.
2. A "closed" linear subspace is also not necesarilly finite but is dense in $$H$$? What is the meaning of the $$s$$ in $$\mathscr{P}_{\mathscr{S}}s$$? What is the difference between closed and finite dimensional?
3. We assume a new inner product in $$\mathscr{S}$$, but... I thought the inner product whas inerhited by $$H$$.

1. Yes, if $$H$$ is infinite-dimensional, it will have infinite-dimensional subspaces.
2. Asserting that a subset $$S$$ of $$H$$ is dense means that $$\overline S=H$$. So, the only subset of $$H$$ which is both closed and dense if $$H$$ itself.
3. Yes, $$\mathscr S$$ inherits an inner product from $$H$$, but nothing prevents you from defining a new one.
• Y sabes que significa la $s$? @José Carlos Santos, también lo que comentas es que la clausura de $S$ no tiene porque ser $H$