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.


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.

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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 enter image description here

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 $x$.

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  • $\begingroup$ but is the right paranthesis miswritten? it doesnt make sense for me why the absolute sign of $a_{ij}$ is outside of it $\endgroup$ – user134489 Sep 16 '14 at 15:09
  • 1
    $\begingroup$ @user134489: Ops, typo. I edited the equation. I would suggest reading any numerical analysis text book for detailed explanation about norms and iterative methods for linear systems. (See "Numarical Analysis, Burden & Faires") $\endgroup$ – Meshal Sep 16 '14 at 16:03

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