Basis for L_infty(R) - MathOverflow

Basis for L_infty(R)

2010-08-09T04:37:13Z

Let $V$ be the Banach space of bounded sequences of reals with the sup norm. Does there exists a subset $B$ of $V$ such that

Linear Independence: For all functions $c$ in $\mathbb{R}^B$, if $\sum_{b \in B} c(b) \cdot b = 0$, then $c$ is identically zero.
Spanning Set: For all vectors $v$ in $V$, there exists a function $c$ in $\mathbb{R}^B$ such that $\sum_{b \in B} c(b) \cdot b = v$.

If so, is an explicit such $B$ known?

Answer by Fedor Petrov for Basis for L_infty(R)

2010-08-09T05:51:29Z

It exists for any linear space (Banach structure is not essential here), is called Hamel base. No explicit construction (without use of Axiom of Choice) exists (and, I guess, it may proved in a sense - that existence of Hamel base in any linear space implies the axiom of choice or smth similar).

Answer by Bill Johnson for Basis for L_infty(R)

2010-08-09T14:27:39Z

The space $\ell^\infty_R$ does not have even an M-basis; i.e., a biorthogonal set $(x_t,x_t^*)$ such that the span of the $x_t$ is dense and the $x_t^*$ are total (Lindenstrauss, late 1960s IIRC), so it has nothing like a Schauder basis. Later I proved [PAMS 26. no. 3 467-468 (1970)] that $\ell^\infty$ also does not have an M-basis. However, each of these spaces does have a biorthogonal set $(x_t,x_t^*)$ such that the span of the $x_t$ is dense. This is in my paper with W.J. Davis [Studia Math. 45 173-179 (1973)].