Exercise 6.1.24

Check the three conditions one by one.

1.

• $$\parallel A\parallel =\underset{i,j}{\max <!--\; FUNCTION\; APPLICATION\; -->}\left|A_{\mathit{ij}}\right|\ge 0,$$

  and the value equals to zero if and only if all entries of $A$ are zero.

• $$\parallel \mathit{aA}\parallel =\underset{i,j}{\max <!--\; FUNCTION\; APPLICATION\; -->}\left|{(\mathit{aA})}_{\mathit{ij}}\right|=\underset{i,j}{\max <!--\; FUNCTION\; APPLICATION\; -->}\left|a\right|\left|A_{\mathit{ij}}\right|$$

  $$=\left|a\right|\underset{i,j}{\max <!--\; FUNCTION\; APPLICATION\; -->}\left|A_{\mathit{ij}}\right|=\left|a\right|\parallel A\parallel .$$

• $$\parallel A+B\parallel =\underset{i,j}{\max <!--\; FUNCTION\; APPLICATION\; -->}\left|{(A+B)}_{\mathit{ij}}\right|=\underset{i,j}{\max <!--\; FUNCTION\; APPLICATION\; -->}|A_{\mathit{ij}}+B_{\mathit{ij}}|$$

  $$\le \underset{i,j}{\max <!--\; FUNCTION\; APPLICATION\; -->}\left|A_{\mathit{ij}}\right|+\underset{i,j}{\max <!--\; FUNCTION\; APPLICATION\; -->}\left|B_{\mathit{ij}}\right|=\parallel A\parallel +\parallel B\parallel .$$

2.

• $$\parallel f\parallel =\underset{t\in [0,1]}{\max <!--\; FUNCTION\; APPLICATION\; -->}|f(t)|\ge 0,$$

  and the value equals to zero if and only if all value of $f$ in $[0,1]$ is zero.

• $$\parallel \mathit{af}\parallel =\underset{t\in [0,1]}{\max <!--\; FUNCTION\; APPLICATION\; -->}|(\mathit{af})(t)|=\underset{t\in [0,1]}{\max <!--\; FUNCTION\; APPLICATION\; -->}\left|a\right||f(t)|$$

  $$=\left|a\right|\underset{t\in [0,1]}{\max <!--\; FUNCTION\; APPLICATION\; -->}|f(t)|=\left|a\right|\parallel f\parallel .$$

• $$\parallel f+g\parallel =\underset{t\in [0,1]}{\max <!--\; FUNCTION\; APPLICATION\; -->}|(f+g)(t)|=\underset{t\in [0,1]}{\max <!--\; FUNCTION\; APPLICATION\; -->}|f(t)+g(t)|$$

  $$\le \underset{t\in [0,1]}{\max <!--\; FUNCTION\; APPLICATION\; -->}|f(t)|+\underset{t\in [0,1]}{\max <!--\; FUNCTION\; APPLICATION\; -->}|g(t)|=\parallel f\parallel +\parallel g\parallel .$$

3.

• $$\parallel f\parallel ={\int \; <!--\; nolimits\; -->}_0^1|f(t)|\mathit{dt}\ge 0,$$

  and the value equals to zero if and only if $f=0$. This fact depend on the continuity and it would be an exercise in the Advanced Calculus coures.

• $$\parallel \mathit{af}\parallel ={\int \; <!--\; nolimits\; -->}_0^1|\mathit{af}(t)|\mathit{dt}={\int \; <!--\; nolimits\; -->}_0^1\left|a\right||f(t)|\mathit{dt}$$

  $$=\left|a\right|{\int \; <!--\; nolimits\; -->}_0^1|f(t)|=\left|a\right|\parallel f\parallel .$$

• $$\parallel f+g\parallel ={\int \; <!--\; nolimits\; -->}_0^1|f(t)+g(t)|\mathit{dt}\le {\int \; <!--\; nolimits\; -->}_0^1|f(t)|+|g(t)|\mathit{dt}$$

  $$={\int \; <!--\; nolimits\; -->}_0^1|f(t)|\mathit{dt}+{\int \; <!--\; nolimits\; -->}_0^1|g(t)|\mathit{dt}=\parallel f\parallel +\parallel g\parallel .$$

4.

• $$\parallel (a,b)\parallel =\max <!--\; FUNCTION\; APPLICATION\; -->\{|a|,|b|\}\ge 0,$$

  and the value equals to zero if and only if both $a$ and $b$ are zero.

• $$\parallel c(a,b)\parallel =\max <!--\; FUNCTION\; APPLICATION\; -->\{|\mathit{ca}|,|\mathit{cb}|\}=\max <!--\; FUNCTION\; APPLICATION\; -->\{|c\left|\right|a|,|c\left|\right|b|\}$$

  $$=\left|c\right|\max <!--\; FUNCTION\; APPLICATION\; -->\{|a|,|b|\}=\left|c\right|\parallel (a,b)\parallel .$$

• $$\parallel (a,b)+(c,d)\parallel =\max <!--\; FUNCTION\; APPLICATION\; -->\{|a+c|,|b+d|\}\le \max <!--\; FUNCTION\; APPLICATION\; -->\{|a|+|c|,|b|+|d|\}$$

  $$\le \max <!--\; FUNCTION\; APPLICATION\; -->\{|a|,|b|\}+\max <!--\; FUNCTION\; APPLICATION\; -->\{|c|,|d|\}=\parallel (a,b)\parallel +\parallel (c,d)\parallel .$$