# Is there a notion of a continuous basis of a Banach space?

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.

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