Useful Tools

Triangle Inequality

For any norm \(\|\cdot\|\), the following inequation holds:

\[\|\boldsymbol{a} + \boldsymbol{b}\| \le \|\boldsymbol{a}\| + \|\boldsymbol{b}\|\]

Cauchy-Schwartz Inequality

\[\left(\boldsymbol{x}^\mathsf{T}\boldsymbol{y}\right)^2 \le \boldsymbol{x}^\mathsf{T}\boldsymbol{x}\; \cdot \;\boldsymbol{y}^\mathsf{T}\boldsymbol{y}\]

\[|\boldsymbol{x}^\mathsf{T}\boldsymbol{y}| \le \|\boldsymbol{x}\|_2 \cdot \|\boldsymbol{y}\|_2.\]