l-p vector space

[[concept]]

Definition

Matrix Analysis

${\ell }^p$ norm

For $K^n$ over $K$ the ${\ell }^p$ norm for $p\in \mathbb{Z}$ is defined as $||x|{|}_p={\left({\sum }_{i=1}^n|x_i{|}^p\right)}^{1/p}$

• When $p=1$ this is the “manhattan norm”
• When $p=2$ this is the euclidean norm
• $||x|{|}_{\infty }=\max (x_i)$

Functional Analysis

${\ell }^p$ space

${\ell }^p=\{a=\{a_j{\}}_{j=1}^{\infty }|||a|{|}_p<\infty \}$ is the space of (infinite) sequences. We define the ${\ell }^p$ norm as

(\sum_{i=1}^\infty \lvert a_{i} \rvert^p)^{1/p}\quad\quad 1 \leq p <\infty \\ \sup_{1 \leq j < \infty} \lvert a_{j} \rvert\qquad\,\,\; p = \infty \end{cases} $$

Example

${\left\{\frac{1}{j}\right\}}_{j=1}^{\infty }\in {\ell }^p\forall p>1$ but not for $p=1$

References

References

See Also

• L-p norm, L-p space
• inner product

Mentions

Mentions

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