I'm reading "The Elements of Statistical Learning" and early on there are references to p-vectors (page 10) and K-vectors (page 12).
What exactly is meant by a p-vector and K-vector?
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It's merely some generic notation for a vector of $p$ attributes or variables observed on $i=1,\dots, N$ individuals, so that you can define $X^T = (X_1,X_2,\dots,X_p)$ as a vector of inputs, in the feature (or input) space (and each individual will have one such vector of observed inputs).
The $K$ notation seems to be reserved to the output space: in a classical linear regression model where $Y=X\beta$, Y is a scalar ($K=1$), whereas in a multivariate setting (say, you record weight, height, and color) it could be a $K$-vector (i.e., 3-vector with my example).
In mathematics and physics, the "x" in "x-vector" stands for the dimension of the vector. The meanings of $K$ and $p$ were previously established. Typically a "p-vector" is written as a column vector and a "p-covector" would be written as a row vector.
So there is a confusion if one is coming from the domain of machine learning.
In the beginning of the book it said that: features = inputs = a set of variables (X1,.... Xp). "inputs" is not related to the number of input samples. Input samples (training samples) are called "observations" (1 .. i .. N).