Linear subspace – closed subspace – new inner product

hilbert-spacesinner-productsvector-spaces

"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 $H$.

Best Answer

  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.
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