the Euclidean norm is denoted using || x ||, right? Then what does || x ||_{∞} mean?

Yes, the Euclidean norm is commonly denoted by \( \|x\| \), and it is specifically the \( \ell_2 \) norm, which is defined for a vector \( x = (x_1, x_2, \dots, x_n) \) as:

\[
\|x\| = \|x\|_2 = \sqrt{x_1^2 + x_2^2 + \dots + x_n^2}
\]

The notation \( \|x\|_{\infty} \) refers to the **infinity norm** (or \( \ell_{\infty} \) norm) of the vector \( x \). It is defined as the maximum absolute value of the components of the vector \( x \). Mathematically, for a vector \( x = (x_1, x_2, \dots, x_n) \), the infinity norm is:

\[
\|x\|_{\infty} = \max_{1 \leq i \leq n} |x_i|
\]

### Do you want details or have any questions?

### Expanding Questions:
1. How does the \( \ell_1 \) norm differ from the \( \ell_2 \) and \( \ell_{\infty} \) norms?
2. Can you explain the geometric interpretation of the \( \ell_{\infty} \) norm?
3. How are the different norms \( \ell_1 \), \( \ell_2 \), and \( \ell_{\infty} \) used in various applications like optimization?
4. What happens to the value of \( \|x\|_{\infty} \) as the dimension of the vector \( x \) increases?
5. How would you compute the \( \ell_{\infty} \) norm for a specific vector, for example, \( x = (3, -5, 2, 7) \)?

### Tip:
Different norms can be used to measure different types of "distances" or "sizes" in vector spaces, depending on the specific requirements of the problem you're dealing with.

Discover the definitions and differences between the Euclidean norm (\( \ell_2 \) norm) and the infinity norm (\( \ell_{\infty} \) norm) in vector spaces. Learn how these norms are calculated and their applications in mathematics and optimization.

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