### Advanced Analysis, Notes 10: Banach spaces (application: divergence of Fourier series)

Recall Theorem 6 from Notes 3:

Theorem 6: For every $f \in C_{per}([0,1]) \cap C^1([0,1])$, the Fourier series of $f$ converges uniformly to $f$

It is natural to ask how much can we weaken the assumptions of the theorem and still have uniform convergence, or how much can we weaken and still have pointwise convergence. Does the Fourier series of a continuous (and periodic) function always converge? In this post we will use the principle of uniform boundedness to see that the answer to this question is a very big NO.

Once again, we begin with some analytical preparations.  Read the rest of this entry »

### Advanced Analysis, Notes 9: Banach spaces (the three big theorems)

Until now we had not yet seen a theorem about Banach spaces — the Hahn–Banach theorems did not require the space to be complete. In this post we learn the three big theorems about operators on Banach spaces: the principle of uniform boundedness, the open mapping theorem, and the closed graph theorem. It is common that these three theorems are presented in texts on functional analysis under the heading “consequences of the Baire category theorem“.  Read the rest of this entry »