Thread: finite spanning set<->frame

1. finite spanning set<->frame

Show that every finite spanning set for $\displaystyle \mathbb{R}^d$ is a frame for $\displaystyle \mathbb{R}^d$
(frame:$\displaystyle \exists A,B >0 \in such that A ||x||^2<=\sum|<x,f_k>|^2<=B||x||^2$ )

and that every finite frame $\displaystyle {f_1,\ldots,f_n}\subset \mathbb{R}^d, n>d$ is a spanning set for $\displaystyle \mathbb{R}^d$

2. finite frame->spanning

Assume $\displaystyle {f_1,\ldots,f_n}$ is a frame, but not a spanning set $\displaystyle \rightarrow$ nontrivial null space for the matrix with rows $\displaystyle f_k\rightarrow\exists x$ such that $\displaystyle ||x||>0$ but $\displaystyle \forall k, |<x,f_k>|=0\rightarrow$ Contradiction of framing condition for any postive A

3. Thanks. Also, half of the other direction is easy:

$\displaystyle \sum |<x,f_k>|^2\leq\sum ||x||^2||f_k||^2$ by Cauchy-Schwartz $\displaystyle =||x||^2\sum||f_k||^2$ and spanning$\displaystyle \rightarrow \exists k$such that$\displaystyle ||f_k||>0\rightarrow B=\sum||f_k||^2$