Interesting Subspaces of ##L^p## Spaces

[nateHI]

$C_0=\{f\in L^p: f(x)\rightarrow 0$ as $x\rightarrow infinity\}$

This is an interesting subspace because it is the subspace of $L^p$ in which the momentum operator from physics is self adjoint. It seems that there should be more to be said about the importance of $C_0$ though. This has probably been studied before but I'm hoping for a summary and possibly some links to relevant papers on the subject that an undergrad in math might be able to read. What other interesting subspaces of $L^p$ are there?

 

[micromass]

$C_0=\{f\in L^p: f(x)\rightarrow 0$ as $x\rightarrow infinity\}$

Hmm, two functions are equal in $L^p$ if they are equal almost everywhere. And just because one function $f(x)\rightarrow 0$ as $x\rightarrow +\infty$, doesn't mean exactly that $g(x)\rightarrow 0$ as $x\rightarrow +\infty$ with $f=g$ a.e. So you need to adjust your subspace a bit. But ok, I know what you mean.

This is an interesting subspace because it is the subspace of $L^p$ in which the momentum operator from physics is self adjoint.

Sure, except that the momentum operator isn't well-defined on entire $C_0$ but only a subspace of it.

It seems that there should be more to be said about the importance of $C_0$ though. This has probably been studied before but I'm hoping for a summary and possibly some links to relevant papers on the subject that an undergrad in math might be able to read. What other interesting subspaces of $L^p$ are there?

If you want important subspaces, then you really can't beat http://en.wikipedia.org/wiki/Schwartz_class

 

[jbunniii]

Another interesting subspace along the lines of the one you pointed out is the Schwartz space, consisting of smooth functions which decrease rapidly to zero (and all of the derivatives have the same property). This is a subspace of $L^p$ (indeed, of your $C_0$) for every $1 \leq p \leq \infty$. It is often used as the set of test functions upon which distributions act:

http://en.wikipedia.org/wiki/Schwartz_space

Another subspace of your $C_0$ is the space of continuous functions with compact support. These functions are identically zero if you take $x$ large enough, so certainly they satisfy your limit.

Then there are the $C^k$ spaces, consisting of functions which are $k$ times differentiable, and whose $k$th derivative is continuous. These may not be properly contained in $L^p$ (it depends on whether you are talking about $L^p(\mathbb{R})$ or $L^p$ on some smaller bounded set such as a finite length interval or the unit circle). But certainly $C^k \cap L^p$ is a subspace of $L^p$ in any case. The $C^\infty$ functions are also called smooth functions. You can also obtain a subspace of $C^k$ by considering the space of $C^k$ functions with compact support.

Similarly, the spaces of polynomials and trigonometric polynomials are extremely useful, and they are subspaces of $L^p$ if the underlying set is bounded.

Also, if $q \neq p$, then $L^p \cap L^q$ can be of considerable interest. For example, we can use a sequence of functions in $L^1 \cap L^2$ to define the Fourier transform of a function in $L^2$, where the usual integral definition may not work.

These are just some examples off the top of my head - there are many other interesting subspaces as well.

 

[jbunniii]

I see micro beat me to the punch regarding the Schwartz space.

 

[micromass]

And then there are of course the Hardy and Sobolev spaces...

 

[micromass]

Also interesting is the Wiener algebra, the Lipschitz functions and the H\"older continuous functions. (Yeah, I'm not working with $\mathbb{R}$ as underlying set, but $S^1$).

 

[micromass]

Or what about the absolute continuous functions? Let's denote them by $AC[0,1]$ if the domain of definition of $[0,1]$. Then for any $\alpha\in \mathbb{C}$ with $|\alpha|=1$, we can define

$$\{\varphi\in AC[0,1]~\vert~\varphi(0) = \alpha\varphi(1)\}$$

These are exactly the domains on which the operator $i\frac{d}{dx}$ is self-adjoint.

 

[nateHI]

Wow, good stuff.

I guess the first thing I need to do work through why $C_0$ isn't the subspace with the least amount of requirements for the momentum operator to be self-adjoint like I thought it was.

Quick question, I thought $C_0$ was first considered by Banach? Am I mistaken about that also?

 

[micromass]

Wow, good stuff.

I guess the first thing I need to do work through why $C_0$ isn't the subspace with the least amount of requirements for the momentum operator to be self-adjoint like I thought it was.

You need to be able to take derivatives if you want the momentum operator to be defined. Is a general function in $C_0$ differentiable?

Quick question, I thought $C_0$ was first considered by Banach? Am I mistaken about that also?

You might be confusing this with the space

$$\mathcal{C_0}(\mathbb{R})$$

which is the set of continuous functions converging to $0$ at infinity. This is indeed quite an important space. It's not a subspace of $L^p$ however. If you want to know more about this very interesting space, then you should be studying C*-algebras.

 

[micromass]

If you want to know the exact subspace of $L^2(\mathbb{R})$ on which the momentum operator $-ih\frac{d}{dx}$ is self-adjoint, then define

$$\mathcal{C}_c^\infty(\mathbb{R})$$

the set of all smooth functions which are only nonzero on some subset of a closed interval.
Now, we say that $f\in L^2(\mathbb{R})$ is distributionally differentiable in $L^2(\mathbb{R})$ iff there exists a $g\in \mathbb{R}^2$ such that for all $\varphi\in C_c^\infty(\mathbb{R})$ holds that

$$\int_{-\infty}^{+\infty} \overline{\varphi^\prime(x)}f(x)dx = -\int_{-\infty}^{+\infty} \overline{g(x)}\varphi(x)dx$$

We denote $f^\prime = g$ and call this the distributional derivative. Now let $A$ be the set of all elements in $L^2(\mathbb{R})$ which are distributionally differentiable in $L^2(\mathbb{R})$, then $-ih\frac{d}{dx}$ is self-adjoint on $A$.

The set $A$ is of course better know as the Sobolev space, which is denoted by $H^1(\mathbb{R})$ or $W^{1,2}(\mathbb{R})$. It becomes a Hilbert space if we equip it with the inner product

$$<f,g> = \int_{-\infty}^{+\infty} (\overline{f(x)}g(x) + \overline{f^\prime(x)}g^\prime(x))dx$$

The normal $L^2$-inner product on $A$ is of course

$$(f,g) = \int_{-\infty}^{+\infty} \overline{f(x)}g(x)dx$$

and then $A$ is not even complete, since it is dense in $L^2(\mathbb{R})$.

The following text should be interesting to you: http://files.vipulnaik.com/exposition/functionspaces.pdf

 

[nateHI]

The set $A$ is of course better know as the Sobolev space, which is denoted by $H^1(\mathbb{R})$ or $W^{1,2}(\mathbb{R})$. It becomes a Hilbert space if we equip it with the inner product

That is extremely interesting indeed. So much to think about now...

 

[nateHI]

The following text should be interesting to you: http://files.vipulnaik.com/exposition/functionspaces.pdf

It was interesting, unfortunately most of it was to advanced for me. I struggled through it but to be honest I had to give up on the topological stuff. I mentioned to my professor that I was reading this and he guaranteed me a grad school recommendation if I wrote a report on the topic of $L^p$ spaces. Funny how these things work out.

 

[micromass]

It was interesting, unfortunately most of it was to advanced for me. I struggled through it but to be honest I had to give up on the topological stuff. I mentioned to my professor that I was reading this and he guaranteed me a grad school recommendation if I wrote a report on the topic of $L^p$ spaces. Funny how these things work out.

That's pretty cool of your professor! What exactly do you need to write a report about? Anything to do with $L^p$ spaces. I can give you some good references if you want. Do you know measure theory?

 

[nateHI]

He said it can be anything to do with $L^p$ spaces as long as I find it interesting. My interest is building an "intuition" for concepts from physics and I feel like I can do that by focusing on the math.

For example, my first introduction to $L^p$ spaces was simply that $iff$ p=2 can you have a Hilbert space and then I went on to learn Quantum Mechanics. But a more interesting question, I feel is, what is an $L^p$ space and will studying subspaces or more general spaces increase my understanding of nature?

I have been teaching myself measure theory but I haven't had a class in it. I've been reading, "An Introduction to Topology and Modern Analysis" by Simmons.

 

[economicsnerd]

My favourite is the (above mentioned) Hardy space $H^p$, which can simultaneously be seen as a subspace of $L^p(\text{the unit circle})$ and a subspace of complex-differentiable functions on the unit disk.