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If $X$ is a Banach space, then a Hamel basis of $X$ is a subset $B$ of $X$ such that every element of $X$ can be written uniquely as a linear combination of elements of $B$. And a Schauder basis of $X$ is a subset $B$ of $X$ such that every element of $X$ can be written uniquely as an infinite linear combination of elements of $B$.

But my question is, is there a notion of a “continuous basis” of a Banach space? That is, a subset $B$ of $X$ such that every element of $X$ can be written uniquely in terms of some kind of integral involving elements of $B$.

I’m not sure what the integral should look like, but one possibility is this. We define some function $f:\mathbb{R}\rightarrow X$, and we let $B$ be the range of $f$. And then for any $x\in X$, there exists a unique function $g:\mathbb{R}\rightarrow\mathbb{R}$ such that $x = \int_{-\infty}^\infty g(t)f(t)dt$, where this is a Bochner integral. And if that’s the case we say that $B$ is a continuous basis for $X$. Does any of this make sense?

EDIT: I've realized that my question is related to a whole bunch of other topics, including Fourier transforms, Rigged Hilbert Spaces, and Spectral Theory. See this answer, this answer, this question, this question, and this question.

Keshav Srinivasan
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  • Some context might help. Every vector space already has a Hamel basis so it's unclear what you hope to gain. – Ben W Apr 05 '19 at 13:47
  • @BenW A Hamel basis involves finite linear combinations. A Schauder basis involves countably infinite linear combinations. I want something that involves “uncountable linear combinations”, i.e. integrals. – Keshav Srinivasan Apr 05 '19 at 14:14
  • @BenW I just made an edit that provides more context. – Keshav Srinivasan Apr 08 '19 at 01:22

2 Answers2

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In the Hilbert case, there is the concept of "rigged Hilbert space", and the one of "generalized eigenvector", which is exactly what you are after.

https://en.wikipedia.org/wiki/Rigged_Hilbert_space

I like the treatment of this on the book of Ballentine on quantum mechanics.

Giuseppe Negro
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The notion of Rigger Hilbert Space mentioned by Giuseppe definitely is a possible answer. For a concrete example if one has a function $f:\mathbb{R}\rightarrow\mathbb{R}$ then one can think of it as a continuous linear combination $$ f=\int_{\mathbb{R}} c_t\ e_t\ dt $$ where $(e_t)_{t\in\mathbb{R}}$ is the "basis" of functions $$ e_t(s)=\delta(s-t) $$ given by Dirac delta's at all possible locations $t$. This is like the continuous analogue of the canonical basis of $\mathbb{R}$ which is the space of functions from $\{1,2,\ldots,n\}$ to $\mathbb{R}$. The coordinates of the function $f$ are just $c_t=f(t)$. The key point to keep in mind for this type of "continuous bases" is that they are made of vectors which are outside the function space one is interested in.

Also note that producing a basis for a space is the same as writing is as direct sum of one-dimensional subspaces. There is a continuous analogue of the notion of direct sum, namely, the notion of direct integral. This could also be another answer for this question.

Abdelmalek Abdesselam
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