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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.

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  • $\begingroup$ Some context might help. Every vector space already has a Hamel basis so it's unclear what you hope to gain. $\endgroup$ – Ben W Apr 5 at 13:47
  • $\begingroup$ @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. $\endgroup$ – Keshav Srinivasan Apr 5 at 14:14
  • $\begingroup$ @BenW I just made an edit that provides more context. $\endgroup$ – Keshav Srinivasan Apr 8 at 1:22

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