Functional Analysis-Two Questions

Question

I am having two questions:

• Does every separable Hilbert Space have a Schauder Basis?

Proof: Every Hilbert Space $H$ has a total orthonormal set say $E=\{e_i\}_{i\in I}$. Since $H$ is separable to $E$ is countable then $E=\{e_k\}_{k\in \Bbb N}$.

Then $E$ is a Schauder Basis of $H$.

Since $E$ is total so $\text{span E}$ is dense in $H$ hence given $x\in H$ there exists a sequence $x_n\to x$ such that $||x_n-x||\to 0$ where $x_n=\sum_{i=1}^m c_{ni}e_i$.

Is it correct?

• I know that every normed space has a Hamel Basis since they are all vector spaces.What is an example of a normed space which has no Schauder Basis?

Answer 1

Your reasoning is correct. Another way of proving this, is to take an countable dense set, and then apply Gram-Schmidt procedure to generate an orthonormal Schauder basis.

As to your second question: If a normed space has a Schauder basis, then the space is separable, since the set of finite linear combinations of basis vectors with rational (complex) coefficients is countable and dense. Hence, non-separable spaces do not admit a Schauder basis.