A bounded linear functional is a type of linear map from a vector space to its underlying field that is continuous and preserves the structure of the vector space. Specifically, it takes a vector and produces a scalar in such a way that the mapping respects addition and scalar multiplication, while also being limited in how large it can get, meaning there is a constant that bounds its output for all input vectors. This concept is crucial in understanding dual spaces and plays a significant role in reproducing kernel Hilbert spaces.
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