All Questions

Let $H$ a Hilbert space, and $\mathcal C$ an arbitrary set of closed subspaces of $H$. Is it true that $$\left( \bigcap_{Z\in \mathcal C}Z\right)^\perp = \overline{\sum_{Z\in \mathcal C} Z^\perp}$$ ...

Is it true that, assuming the Axiom of Choice, every infinite-dimensional Banach space has an infinite-dimensional closed subspace with infinite codimension? Note that this is different from the ...

A subset of a linear space $X$ is called infinite-dimensional if it is not contained in a finite-dimensional linear subspace of $X$. Problem. Let $L$ be an infinite-dimensional subset of the linear ...

Let $k$ be a field and $V$ a $k$-vector space. Then there is a map $V \to V^{\ast \ast}$, where $V^{\ast}$ is the dual vector space. If we are in ZFC and $\dim V$ is infinite, then this map is not ...