What does$ \|x\|_{\infty}$ mean?

Question

In one of my homework-assignments in analysis, I have stumpled upon $ \|x\|_{\infty}$. I know x is a vector, but what does the infinity-symbol imply? The whole problem is actually this:

$\| Ax\|_{\infty} \leq |\max_{1 \leq i \leq n}(\Sigma_{j=1}^n \left| a_{ij}) \right| \|x\|_{\infty}$

Im also not 100% sure what the infinity-sign to the left also mean, so it would be nice if anyone could elaborate. Does the way the right paranthesis is presented now make sense, or is it a clear error in the problem set?

Thanks for your help.

Answer 1

It denotes the maximum (or, in case of sequence or function spaces, the supremum) norm (on both sides of the inequality),

$$\lVert x\rVert_\infty = \max \{ \lvert x_k\rvert : 1 \leqslant k \leqslant n\}$$

if $x\in \mathbb{R}^n$ or $x\in\mathbb{C}^n$.

For $1 \leqslant p < \infty$, one has the norms

$$\lVert x\rVert_p = \left(\sum_{k=1}^n \lvert x_k\rvert^p\right)^{1/p},$$

and for every $x$ one has

$$\lVert x\rVert_\infty = \lim_{p\to\infty} \lVert x\rVert_p,$$

which may explain the notation.

Answer 2

Mathematically a norm is a total size or length of all vectors in a vector space or matrices. And by definition,

$$\left \| x \right \|_n=\sqrt[n]{\sum _i\left | x_i \right |^{n}}, n\in \mathbb{R}$$

Now let us denote the highest entry in the vector $x$ by $x_j$ (assume it is the $j$-th entry).

Thus we we wolud have,

$${\sum _ix_i^{\infty }}\cong x_j^{\infty }$$

So using the definition above, we get the important result,

$$\left \| x \right \|_\infty =\sqrt[\infty ]{\sum _ix_i^{\infty }}=\sqrt[\infty ]{x_j^{\infty }}=\left | x_j \right |=max(\left | x_i \right |)$$

---

• What you have written above

simply means that the maximum (in absolute value) component of the vector $Ax$ is less than or equal of the maximum component in the matrix $A$ multiplied by the maximum component of the vector $x$.