# Representing a Banach space as a function space

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 $$. And I'm "always" right, I can even guess properties of the function space $$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$$".

• 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. Dec 31, 2021 at 13:12

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 $$p < \infty$$.
• 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. Nov 13, 2018 at 1:47
• 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. Nov 13, 2018 at 11:05
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 $$x \in X$$.