## Month: November, 2012

### 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 »

### On the isomorphism question for complete Pick multiplier algebras

In this post I want to tell you about our new preprint, “On the isomorphism question for complete Pick multiplier algebras“,  which Matt Kerr, John McCarthy and myself just uploaded to the arXiv. Very broadly speaking, the motif of this paper is the connection between algebra and geometry; to be a little bit more precise, it is the connection between complex geometry and nonself-adjoint operator algebras. Read the rest of this entry »

### Advanced Analysis, Notes 8: Banach spaces (application: weak solutions to PDEs)

Today I will show you an application of the Hahn-Banach Theorem to partial differential equations (PDEs). I learned this application in a seminar in functional analysis, run by Haim Brezis, that I was fortunate to attend in the spring of 2008 at the Technion.

As often happens with serious applications of functional analysis, there is some preparatory material to go over, namely, weak solutions to PDEs.

### Advanced Analysis, Notes 7: Banach spaces (dual spaces and duality, Lp spaces, the double dual, quotient spaces)

Today we continue our treatment of the dual space $X^*$ of a normed space (usually Banach) $X$. We start by considering a wide class of Banach spaces and their duals.  Read the rest of this entry »