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Questions about the properties of vector spaces and linear transformations, including linear systems in general.

1 vote
Accepted

If a vector space has a basis then its dual vector space has a basis

Here are two statements you might find interesting: With base field $k\subseteq\mathbb C$ your axiom implies that any free $\mathbb Z$-action has a choice of representatives. For each prime $p,$ ther …
Harry West's user avatar
  • 1,916
11 votes
Accepted

Existence of translation-invariant basis on $C_c(\mathbb R)$

No, there is no shift-invariant basis of $V=C_c(\mathbb R).$ I'll use the formulation in YCor's answer, so we need to show that $V$ is not a free $B$-module where $B=\mathbb C[T^r:r\in \mathbb R],$ wi …
Gro-Tsen's user avatar
  • 32.5k
3 votes
Accepted

Example of an inverse system which suddenly "jumps" in size in a specific "controlled" way?

Here is my argument, which assumes $|2^\omega|<|2^{\omega_1}|$ and uses an infinite base field. See Tim Campion's answer https://mathoverflow.net/a/376790/164965 for the general case. Take the base fi …
Harry West's user avatar
  • 1,916