Statistics and machine learning references which require functional analysis background I am doing a PhD in functional analysis, particularly Banach space theory. Thinking of my future prospect, I would like to venture myself into statistics. 
However, as I do not want to 'waste' my background, I would like to search for statistics or machine learning which requires functional analysis to understand its underlying concepts. 
I found Bickel et al. Efficient and Adaptive Estimation for Semiparametric Models, Tsiatis's Semiparametric Theory and Missing Data and Small et al. Hilbert Space Methods in Probability and Statistical Inference.
Note that I know functional analysis is required for stochastic calculus, theoretical probability, browninan motion, etc. But I am more interested in statistics. 
I also found that similar question was posted reddit.
It would be good if someone can suggest some statistics books that require functional analysis background.
 A: I would suggest looking through the functional analysis literature for papers relating to statistics and machine learning. Books aren't where the cutting-edge results are found, and your background will allow you to understand the functional analysis papers.
In a few minutes on the website for what appears to be a strong functional analysis journal, I found the following papers.
Newton, Nigel J. "An infinite-dimensional statistical manifold modelled on Hilbert space." Journal of Functional Analysis 263.6 (2012): 1661-1681. https://www.sciencedirect.com/science/article/pii/S0022123612002376
Jenčová, Anna. "A construction of a nonparametric quantum information manifold." Journal of Functional Analysis 239.1 (2006): 1-20. https://www.sciencedirect.com/science/article/pii/S0022123606000644
Hangelbroek, Thomas, and Amos Ron. "Nonlinear approximation using Gaussian kernels." Journal of Functional Analysis 259.1 (2010): 203-219. https://www.sciencedirect.com/science/article/pii/S0022123610000467
A: A graduate level book on Statistics and Functional Analysis.  This post Is functional analysis and hilbert spaces useful in machine learning? If so, how?  gives some hints&references. 
A: Simovici 2018, Mathematical Analysis for Machine Learning and Data Mining
This book might be subtitled "an introduction to machine learning for functional analysts".
A: (Not enough reputation to comment.)
Did you take a look at functional data analysis (FDA)? It's a popular branch of statistics. The introductory textbooks probably don't require a heavy math background, but the research done in FDA can be math heavy (if you want it to be).
