Summary: In this paper we are concerned with reproducing kernel Hilbert spaces $$H_{K}$$ of functions from an input space into a Hilbert space $$Y$$, an environment appropriate for multi-task learning. The reproducing kernel $$K$$ associated to $$H_{K}$$ has its values as operators on $$Y$$. Our primary goal here is to derive conditions which ensure that the kernel $$K$$ is universal. This means that on every compact subset of the input space, every continuous function with values in $$Y$$ can be uniformly approximated by sections of the kernel. We provide various characterizations of universal kernels and highlight them with several concrete examples of some practical importance. Our analysis uses basic principles of functional analysis and especially the useful notion of vector measures which we describe in sufficient detail to clarify our results.

### MSC:

 68T05 Learning and adaptive systems in artificial intelligence
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