7
$\begingroup$

When I think about a vector space, I like to see them as spaces of parameters. Although finite dimensional spaces have been generalized to Banach spaces, I see function spaces as the best generalization, at least heuristically, for a space of parameters. Given the function space $L^p(X,\Omega,\mu)$, for simplicity $L^p(X)$, the set of Dirac's deltas at each point works as a kind of canonical base, so when we choose a function, we are specifying somehow $|X|$ number of parameters; $|X|$ is the cardinality of $X$. My first question is: what are sufficient and necessaries conditions for a Banach space to be a function space?

To be more precise:

  1. When is a Banach space $B$ isometric to some space $L^p(X)$? (isometric representation)
  2. When is there a linear isomorphism between $B$ and some $L^p(X)$, that is, a linear operator $A:B\to L^p(X)$ such that $c\lVert x\rVert_B\le \lVert Ax\rVert_{L^p}\le C\lVert x\rVert_B$? (continuous representation)
  3. When is there a space $X$ and a continuous operator $A:C_c(X)\to B$, such that $A$ is 1-1 and the image of $A$ is dense? $C_c(X)$ is the space of compactly supported functions on $X$; we can endow it with the usual topology. (weak representation)

The definition I like the most is the third, I think it's the weaker. The first two definitions involve $L^p$ spaces, and they are not the sole function spaces out there, so these definitions are incomplete. Can you imagine a weaker way of representing a Banach space?

If $B$ is a Hilbert space, take an orthonormal expansion, so every vector is $f=\sum_{i\in I} f(i)e_i$ and we have an isometric representation with $L^2(I)$, but this is not the most appealing representation, and it leaves more questions. What can we say about $X$?

If you take $L^2(\mathbb{R}^n)$, you can use a partition of unity and Fourier series to write every function $f$ as a sum $\sum_{i\in I} f(i)e_i$, where $I$ is countable and the $e_i$ form an orthonormal base. But, what is then the difference between $L^2(\mathbb{R}^2)$ and $L^2(\mathbb{R})$? Doesn't the structure of the space $X$ matter? I feel uneasy.

Many times in my work I have encountered operators like $A:L^p(X)\to L^q(Y)$, where $X$ and $Y$ are topological spaces of dimensions $N$ and $M$, respectively; for simplicity, suppose that $A$ is injective. Heuristically I apply the finite dimensional logic, for example take $N=M$, and I think: the cokernel of $A$ is like a "tiny" function space, compared with $X$ or $Y$, over a topological space $Z$ of dimension $<M$. And I'm "always" right, I can even guess properties of the function space $Z$, like $\text{dim}(Z)$, from topological properties of $X$ and $Y$. Then, what does it happen with $L^2(\mathbb{R}^n)$? Can we, under additional assumption, prove a kind of invariance of domain theorem for function spaces? It may help the reader to think about the operator $\iota:C_c(\mathbb{R})\to C_c({\mathbb{R}^2})$, given by $\iota f(x,y)\mapsto f(x)\varphi(y)$, where $\varphi$ is supported near to zero, and the quotient $C_c({\mathbb{R}^2})/\iota C_c(\mathbb{R})$; we can think that the dimension of the cokernel is "$\mathbb{R}^2-\mathbb{R}\approx \mathbb{R}^2$".

$\endgroup$
1
  • $\begingroup$ What do you mean by Dirac's deltas? If the measure space is not discrete, the elements of $L^p(X)$ are equivalence classes of functions. $\endgroup$
    – Jochen
    Dec 31, 2021 at 13:12

2 Answers 2

2
$\begingroup$

The third question always has a positive answer. Let $E$ be a Banach space and let $B_{E^*$}$ be the unit ball of the dual space $E^*$ endowed with the weak-$\ast$ continuous topology. Then, any $x \in E$ has a representation as a continuous function $\hat{x} \in C(B_{E^*})$, given by $\hat{x}(\lambda) = \lambda(x)$. Recall that $B_X$ is compact by Banach-Alaouglu.

I do not known a definitive answer to the other questions (identify the closed subspaces of $L^p$). Using Rademacher techniques/Khintchine inequalities you can definitely embed $\ell^2$ inside $L^p$ for $p < \infty$.

$\endgroup$
2
  • $\begingroup$ I do like this, but I have a question: given $x\in E$, the correspoding continuous function is "linear": $\hat{x}(\lambda + \mu)=\hat{x}(\lambda)+\hat{x}(\mu)$, whenever $\lambda,\mu,\lambda+\mu\in B_{E^*}$, hence not every continuous function comes from a vector $x\in E$ in this representation. In the representation I look for, every continuous function with compact support lies inside $E$, in particular, functions approximating any Dirac's delta. Am I right or confused? Thank you. $\endgroup$
    – user90189
    Nov 13, 2018 at 1:47
  • $\begingroup$ You are right, $E$ is a proper subspace of $C(B_E)$. I do not know if there is characterization of Banach spaces which contain the compactly supported functions densely. $\endgroup$ Nov 13, 2018 at 11:05
0
$\begingroup$

One known way of representing any real Banach space as a function space is as follows: Let $X$ be a real Banach space. Let $A(B_{X^{\ast}})$ denote the Banach space of all real-valued affine weak$^{\ast}$-continuous functions on the closed unit ball of $X^{\ast}$, denoted by $B_{X^{\ast}}$, where $B_{X^{\ast}}$ is endowed with the weak$^{\ast}$ topology. Now consider a subspace of $A(B_{X^{\ast}})$ namely $A_{\sigma}(B_{X^{\ast}}) = \{a \in A(B_{X^{\ast}}): a(x^{\ast}) = -a(-x^{\ast})\mbox{ for each }x^{\ast} \in B_{X^{\ast}}\}$. Then $X$ is isometric and linearly isomorphic to $A_{\sigma}(B_{X^{\ast}})$ under the mapping $x \mapsto \hat{x}$, where $\hat{x}(x^{\ast}) = x^{\ast}(x)$, for each $x^{\ast} \in B_{X^{\ast}}$ and $x \in X$.
The reference for this result is the book The Isometric Theory of Classical Banach spaces by H. E. Lacey. https://3lib.net/book/1304350/ffc878

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.