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Infinite-dimensional space

As explained in Bayesian Optimization in Action

An infinite-dimensional space is a mathematical space requiring infinitely many coordinates or basis elements to represent its elements.

An infinite-dimensional space models elements with infinitely many variables. In Gaussian-process modeling, increasing the variables from coarse California regions toward infinitely fine regions lets predictions be made anywhere on the map. section 2.3

The connection to Gaussian processes is defined through finite subsets:

  • A Gaussian process is a collection of random variables whose every finite subset has a multivariate normal distribution. section 2.3
  • Function values such as [f(1), f(2), f(3)] form finite-dimensional Gaussian vectors, while all values together describe a distribution over functions. section 2.3

How Bayesian Optimization in Action covers it

How deep it goes
Defines the concept through finite multivariate-normal subsets and contrasts vectors with distributions over functions.
Examples
Uses California housing regions and increasingly dense sets of function values to illustrate the transition from finite variables to infinitely many.
What you can do after
Evaluate/understand why Gaussian processes model functions through finite-dimensional Gaussian distributions.
How it connects
Connects infinite-dimensional spaces to Gaussian processes, multivariate normal distributions, functions, mean functions, and covariance matrices.
Cover of Bayesian Optimization in Action
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