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Version 12 |
| Let $\mathbb{F}$ be either $\mathbb{R}$ or $\mathbb{C}$, and let $p\in\mathbb{R}$ with $p\geq 1$. We define $\ell^p$ to be the vector space of all sequences $(a_i)_{i\geq 0}$ in $\mathbb{F}$ such that $$\sum_{i=0}^{\infty}|a_i|^p$$ converges, with addition and scalar multiplication defined componentwise. |
Let $\mathbb{F}$ be either $\mathbb{R}$ or $\mathbb{C}$, and let $p\in\mathbb{R}$ with $p\geq 1$. We define $\ell^p$ to be the vector space of all sequences $(a_i)_{i\geq 0}$ in $\mathbb{F}$ such that $$\sum_{i=0}^{\infty}|a_i|^p$$ converges, with addition and scalar multiplication defined componentwise. |
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We can make $\ell^p$ into a normed vector space, by defining the norm as $$\Vert (a_i)\Vert_p = (\sum_{i=0}^{\infty}|a_i|^p)^{1/p}$$.
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We can make $\ell^p$ into a normed vector space, by defining the norm as $$\Vert (a_i)\Vert_p = (\sum_{i=0}^{\infty}|a_i|^p)^{1/p}$$ |
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We also define $\ell^{\infty}$ to be the vector space of all \PMlinkname{bounded}{BoundedInterval} sequences $(a_i)_{i\geq 0}$ with norm given by $$\Vert (a_i)\Vert_{\infty} = \operatorname{sup}\{ |a_i|:i\geq 0\}$$.
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We also define $\ell^{\infty}$ to be the vector space of all \PMlinkname{bounded}{BoundedInterval} sequences $(a_i)_{i\geq 0}$ with norm given by $$\Vert (a_i)\Vert_{\infty} = \operatorname{sup}\{ |a_i|:i\geq 0\}$$ |
| The normed vector spaces $\ell^{\infty}$ and $\ell^p$ for $p\geq 1$ are complete under these norms, making them into Banach spaces. Moreover, $\ell^2$ is a Hilbert space under the inner product $$\langle (a_i),(b_i)\rangle = \sum_{i=0}^{\infty}a_i \overline{b_i}$$ where $\overline{x}$ denotes the complex conjugate of $x$. |
The normed vector spaces $\ell^{\infty}$ and $\ell^p$ for $p\geq 1$ are complete under these norms, making them into Banach spaces. Moreover, $\ell^2$ is a Hilbert space under the inner product $$\langle (a_i),(b_i)\rangle = \sum_{i=0}^{\infty}a_i \overline{b_i}$$ where $\overline{x}$ denotes the complex conjugate of $x$. |
| For $p>1$ the (continuous) dual space of $\ell^p$ is $\ell^q$ where $\frac{1}{p} + \frac{1}{q}=1$, and the dual space of $\ell^1$ is $\ell^{\infty}$. |
For $p>1$ the (continuous) dual space of $\ell^p$ is $\ell^q$ where $\frac{1}{p} + \frac{1}{q}=1$, and the dual space of $\ell^1$ is $\ell^{\infty}$. |
