Complete sequences in $C[0,1]$ and $L^2[0,1]$ It is problem 1.5.4 in Robert M. Young's book, An introduction to nonharmonic Fourier series:
Show that if $\{f_1,f_2,\dots\}$ is a sequence in $C[0,1]$ that is complete in $L^2[0,1]$, then $\{1, f_1, f_2,\dots\}$ is complete in $C[0,1]$.
Completeness here means the set of finite linear combinations is dense.
 A: This is false - is the problem in the book exactly the same as your question?
Say $e_k(t)=e^{ikt}$, and let $(f_n)$ be the sequence $(1,e_1,e_{-1},e_2,e_{-2}\dots)$. Then $(f_n)$ is certainly complete in $L^2([0,1])$. But if $f$ is a linear combination of $(1,f_1,f_2,\dots)$ then $f(1)=f(0)$,  and functions with this property are not dense in $C([0,1])$.
Regardless of exactly how the problem is stated in the book I conjecture that what the author really had in mind was  $L^2(\mathbb T)$ and $C(\mathbb T)$. It's false in that context as well, but the example is not so obvious:
A book on nonharmonic Fourier series probaby contains a proof of this:


Suppose $(f_n)$ is a complete orthonormal sequence in a Hilbert space $H$ and $\sum||f_n-g_n||^2<1$. Then $(g_n)$ is also complete in $H$.


(Sketch of proof: Define $T:H\to H$ by $T\sum a_nf_n=\sum a_ng_n$. Then $||T-I||<1$, hence $T$ is invertible.)
Now let $a$ and $b$ be two distinct points of $\mathbb T$. Let $f_n$ be as above; choose $g_n\in C(\mathbb T)$ so that $g_n(a)=g_n(b)$ and $\sum||f_n-g_n||_2^2<1$. Then $(g_n)$ is complete in $L^2(\mathbb T)$ but $(1,g_1,\dots)$ is not complete in $C(\mathbb T)$.
