Re: Closure of span as infinite sums?

In article <1140422571.081619.322730@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
David <david.davidr@xxxxxxxxx> wrote:

Great -- thanks for this example!
I often come across situations where I wish I could write a generic
element of a set or space in a simple form (like an infinite sum), so
I was just wondering if there were any non-obvious situations when it
would actually be true.


It's called a "Shauder basis" when every element can be written uniquely
as an infinite linear combination.

Another counterexample:
In the space C[0,1] of continuous functions on [0,1] with sup norm,
the polynomials are dense (Weierstrass). So the closed span of the
functions 1, x, x^2, x^3, ... is the whole space. But not every
continuous function is the sum of a power series---for example, such
a sum is differentiable.

The space C[0,1] however has a Schauder basis. Shauder himself
constructed one consisting of certain piecewise linear functions.

All of the commonly-seen separable Banach spaces have Schauder bases.
But (1973) Per Enflo constructed an example of a separable
Banach space with no basis.

--
G. A. Edgar http://www.math.ohio-state.edu/~edgar/
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