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Prove some properties of two alternative definitions of the norm

Consider the following two definitions of the norm of a vector A = (a_1, \ldots, a_n) in \mathbb{R}^n.

    \[ \lVert A \rVert_1 = \sum_{k=1}^n |a_k|, \qquad \text{and} \qquad \lVert A \rVert_2 = \max_{1 \leq k \leq n} |a_k|. \]

Prove that we have the following inequalities for any vector A,

    \[ \lVert A \rVert_2 \leq \lVert A \rVert \leq \lVert A \rVert_1. \]

Give a geometric interpretation of this in the case that n = 2.


Proof. First, for the inequality on the left we have

    \begin{align*}  \lVert A \rVert_2 &= \max_{1 \leq k \leq n} |a_k| \\  &= \max_{1 \leq k \leq n} \left( a_k^2 \right)^2 \\  &\leq \sum_{k=1}^n (a_k^2)^{\frac{1}{2}} \\  &= \lVert A \rVert. \end{align*}

For the inequality on the right, consider

    \begin{align*}  \left(\sqrt{a_1^2 + a_2^2} \right)^2 &= a_1^2 + a_2^2 \\  &\leq a_1^2 + a_2^2 + 2 \sqrt{a_1^2}\sqrt{a_2^2} \\  &= \left( \sqrt{a_1^2} +\sqrt{a_2^2} \right)^2 \end{align*}

Taking square roots of both sides this gives us the inequality

    \[ \sqrt{a_1^2 + a_2^2} \leq \sqrt{a_1^2} + \sqrt{a_2^2}. \]

By induction we can then establish

    \[ \sqrt{a_1^2+\cdots+a_n^2} \leq \sqrt{a_1^2} + \cdots + \sqrt{a_n^2}. \]

Therefore,

    \begin{align*}  \lVert A \rVert &= \left( \sum_{k=1}^n a_k^2 \right)^{\frac{1}{2}} \\[9pt]  &\leq \sum_{k=1}^n \sqrt{a_k^2} \\[9pt]  &= \sum_{k=1}^n |a_k| \\[9pt]  &= \lVert A \rVert_1. \qquad \blacksquare \end{align*}

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