I am reading through Hands-On Machine Learning with Scikit-Learn, Keras, and TensorFlow, by Aurélien Géron.
Chapter 2 he introduces the idea of a general vector norm as follows:
"More generally, the $ℓ_k$ norm of a vector v containing n elements is defined as $∥v∥_k = (|v_0|^k + |v_1|^k + ... + |v_n|^k)^{1/k}$."
My confusion is about what comes next:
"$ℓ_0$ gives the number of nonzero elements in the vector, and $ℓ_∞$ gives the maximum absolute value in the vector.".
I understand how in the limit k going to infinity, the expression will return the absolute value of the vector whos magnitude was greatest, but I'm not sure how in the limit of k tending to 0, the expression returns the number of non zero vectors.
You write $\|v\|^k$ on the l.h.s but on the r.h.s you have $\|v\|_k$, according to the definition of the book you mentioned. The limit of this expression as $k\to 0$ is certainly not equal to the number of non-zero elements of the vector. For example, $(1^k+1^k)^{1/k}$ tends to infinity as $k\to 0^+$, not to $2$. What is true is that $\|v\|_k^k$ tends to the number of the non-zero elements as $k\to 0^+$. The book's comment is therefore misleading.