The max norm, also known as the $$L^\infty$$ norm, measures the absolute value of the largest element in a vector. It is a special case of the $$L^p$$ norm, obtained in the limit as $$p \to \infty$$. For a vector $$\mathbf{x} \in \mathbb{R}^n$$, the max norm is formally defined as: $$\left\|\mathbf{x}\right\|_\infty = \max_{i} |x_i|$$

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The $$L^p$$ norm (or $$\ell_p$$ norm) is a mathematical function that measures the size or length of a vector. For a vector $$\mathbf{x} \in \mathbb{R}^n$$ and a real number $$p \geq 1$$, the $$L^p$$ norm is defined as: $$\left\|\mathbf{x}\right\|_p = \left( \sum_{i=1}^n |x_i|^p \right)^{1/p}$$ Common vector norms are special cases of the $$L^p$$ norm: - The $$L_1$$ norm is obtained by setting $$p=1$$. - The $$L_2$$ norm (Euclidean norm) is obtained by setting $$p=2$$. - The max norm ($$L^\infty$$ norm) is obtained in the limit as $$p \to \infty$$, resulting in: $$\left\|\mathbf{x}\right\|_\infty = \max_{i} |x_i|$$

Lp Norm

Goodfellow, I., Bengio, Y., & Courville, A. (2016). $\mathit{Deep \ Learning.}$ MIT Press. Retrieved from [www.deeplearningbook.org](https://www.deeplearningbook.org) 

Deep Learning

The $$L_1$$ norm, also known as the Manhattan norm or Taxicab norm, measures the sum of the absolute values of a vector's elements. For a vector $$\mathbf{x} \in \mathbb{R}^n$$, the $$L_1$$ norm is defined as:
$$\|\mathbf{x}\|_1 = \sum_{i=1}^n |x_i|$$

L1 Norm

The $$\ell_2$$ norm, also known as the Euclidean norm, measures the Euclidean length of a vector. It is calculated as the square root of the sum of the squares of the vector's elements. For a vector $$\mathbf{x}$$ with $$n$$ elements, the $$\ell_2$$ norm is formally expressed as:
$$\left\|\mathbf{x}\right\|_2 = \sqrt{\sum_{i=1}^n x_i^2}$$

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