███████ █████

██████████ ███████: █████████ ██████ █████

███ of the ████ ████████ ████████ of a ███████ █████ is the █████████ ██████ █████ ██████████ of █████-███████████ ███████, ███████ by ${\displaystyle {\mathbf{<span\; word-length="1"\; class="word\; redacted">█</span>}}^{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}}$, and ████████ with the ███ ███████. The ███ ███████ █████ ███ ███████ █ and █, and ████████ a ████ ██████ █ ⋅ █. If █ and █ are ███████████ in █████████ ███████████, ████ the ███ ███████ is ███████ by: $${\displaystyle \left(\begin{array}{c}{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}_{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}\\ {\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}_{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}\\ {\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}_{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}\end{array}\right)\cdot \left(\begin{array}{c}{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}_{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}\\ {\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}_{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}\\ {\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}_{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}\end{array}\right)={\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}_{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}_{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}+{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}_{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}_{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}+{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}_{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}_{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}.}$$

The ███ ███████ █████████ the ██████████:

1. It is █████████ in █ and █: █ ⋅ █ = █ ⋅ █.
2. It is ██████ in ███ █████ ████████: (a█_█ + ██_█) ⋅ █ = a(█_█ ⋅ █) + █(█_█ ⋅ █) for ███ ███████ a, █, and ███████ █_█, █_█, and █.
3. It is ████████ ████████: for ███ ███████ █, █ ⋅ █ ≥ █ , with ████████ if and ████ if █ = █.

An █████████ on █████ of ███████ that, like the ███ ███████, █████████ █████ █████ ██████████ is █████ as a (████) █████ ███████. A ██████ █████ ████████ with ████ an █████ ███████ is █████ as a (████) █████ ███████ █████. █████ ██████-███████████ █████ ███████ █████ is ████ a ███████ █████. The █████ ███████ of the ███ ███████ that ████████ it with █████████ ████████ is that it is ███████ to ████ the ██████ (or ████) of a ██████, ███████ ‖█‖, and to the █████ █ between ███ ███████ █ and █ by █████ of the ███████ $${\displaystyle \mathbf{<span\; word-length="1"\; class="word\; redacted">█</span>}\cdot \mathbf{<span\; word-length="1"\; class="word\; redacted">█</span>}=\Vert \mathbf{<span\; word-length="1"\; class="word\; redacted">█</span>}\Vert \Vert \mathbf{<span\; word-length="1"\; class="word\; redacted">█</span>}\Vert \mathrm{<span\; word-length="3"\; class="word\; redacted">███</span>}\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}.}$$

█████████████ ████████ in █████████ █████ ██████ on the ███████ to ███████ ██████, and to have ██████ ████████ for ██████████ that ██████ █████. A ████████████ ██████ $${\displaystyle \sum _{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}=\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}^{\mathrm{\infty }}{\mathbf{<span\; word-length="1"\; class="word\; redacted">█</span>}}_{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}}$$ ██████████ of ███████ in █^█ is ██████████ ██████████ ████████ that the ███ of the ███████ █████████ as an ████████ ██████ of ████ ███████: ████ as with a ██████ of ███████, a ██████ of ███████ that █████████ ██████████ ████ █████████ to some █████ ██████ █ in the █████████ █████, in the █████ that $${\displaystyle \underset{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}\to \mathrm{\infty }}{<span\; word-length="3"\; class="word\; redacted">███</span>}\Vert \mathbf{<span\; word-length="1"\; class="word\; redacted">█</span>}-\sum _{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}=\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}^{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}{\mathbf{<span\; word-length="1"\; class="word\; redacted">█</span>}}_{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}\Vert =\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>.}}$$ This ████████ █████████ the ████████████ of █████████ █████: that a ██████ that █████████ ██████████ ████ █████████ in the ████████ █████.

███████ ██████ are █████ █████ over the ███████ ███████. The ███████ █████ ███████ by █ is ████████ with a ██████ of █████████, the ███████ ███████ |█|, which is ███████ as the ██████ ████ of the ███████ of █ with ███ ███████ █████████: $${\displaystyle |\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}{|}^{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}=\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}\overline{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}.}$$

If █ = █ + ██ is a █████████████ of █ into ███ ████ and █████████ █████, ████ the ███████ is the █████ █████████ ███-███████████ ██████: $${\displaystyle |\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}|=\sqrt{{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}^{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}+{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}^{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}}.}$$

The █████ ███████ of a ████ of ███████ ███████ █ and █ is the ███████ of █ with the ███████ █████████ of █: $${\displaystyle ⟨\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>},\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}⟩=\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}\overline{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}.}$$

This is ███████-██████. The ████ ████ of ⟨█, █⟩ █████ the █████ ███-███████████ █████████ ███ ███████.

A ██████ ███████ is the █████ █^█ █████ ████████ are █████ of ███████ ███████ █ = (█_█, █_█). ████ an █████ ███████ of █ with ███████ ████ ██████ █ = (█_█, █_█) is █████ by $${\displaystyle ⟨\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>},\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}⟩={\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}_{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}{\overline{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}}_{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}+{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}_{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}{\overline{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}}_{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}.}$$

The ████ ████ of ⟨█, █⟩ is ████ the ████-███████████ █████████ ███ ███████. This █████ ███████ is █████████ █████████, which █████ that the ██████ of █████████████ █ and █ is the ███████ █████████: $${\displaystyle ⟨\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>},\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}⟩=\overline{⟨\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>},\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}⟩}.}$$

██████████

A ███████ █████ is a ████ or ███████ █████ ███████ █████ that is ████ a ████████ ██████ █████ with ███████ to the ████████ ████████ ███████ by the █████ ███████.

To ███ that a ███████ ██████ █████ █ is a ███████ █████ ███████ █████ █████ that there is an █████ ███████ ${\displaystyle ⟨\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>},\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}⟩}$ ███████████ a ███████ ██████ to ████ ████ of ████████ ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>},\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}$ of █ that █████████ the following ██████████:

1. The █████ ███████ is █████████ █████████; that is, the █████ ███████ of a ████ of ████████ is █████ to the ███████ █████████ of the █████ ███████ of the ███████ ████████: $${\displaystyle ⟨\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>},\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}⟩=\overline{⟨\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>},\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}⟩}.}$$ ███████████, this ███████ that ${\displaystyle ⟨\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>},\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}⟩}$ is a ████ ██████.
2. The █████ ███████ is ██████ in ███ █████ ████████. For ███ ███████ ███████ ${\displaystyle a}$ and ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>},}$ $${\displaystyle ⟨a{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}_{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}+\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}_{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}},\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}⟩=a⟨{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}_{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}},\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}⟩+\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}⟨{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}_{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}},\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}⟩.}$$
3. The █████ ███████ of an ███████ with ██████ is ████████ ████████:

It ███████ from ██████████ █ and █ that a ███████ █████ ███████ is ██████████, ████ ██████ █████████ ██████, in ███ ██████ ████████, ███████ that $${\displaystyle ⟨\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>},a{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}_{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}+\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}_{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}⟩=\overline{a}⟨\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>},{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}_{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}⟩+\overline{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}⟨\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>},{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}_{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}⟩.}$$

A ████ █████ ███████ █████ is ███████ in the ████ ███, except that █ is a ████ ██████ █████ and the █████ ███████ █████ ████ ██████. ████ an █████ ███████ will be a ████████ ███ and ${\displaystyle (\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>},\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>},⟨\cdot ,\cdot ⟩)}$ will ████ a ████ ██████.

The ████ is the ████-██████ ████████ $${\displaystyle \Vert \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}\Vert =\sqrt{⟨\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>},\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}⟩},}$$ and the ████████ ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}$ between ███ ██████ ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>},\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}$ in █ is ███████ in █████ of the ████ by $${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}(\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>},\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>})=\Vert \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}-\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}\Vert =\sqrt{⟨\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}-\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>},\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}-\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}⟩}.}$$ ████, ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}(\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>},\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>})}$ is a ████████ ████████ ███████ ███████ that it is █████████ in ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}$ and ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>},}$ ████████ that the ████████ between ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}$ and ██████ is ████, and █████████ the ████████ between ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}$ and ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}$ ████ be ████████, and ██████ that the ████████ ██████████ █████, ███████ that the ██████ of ███ ███ of a ████████ ███ ██████ ██████ the ███ of the ███████ of the █████ ███ ████: $${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}(\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>},\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>})\le \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}(\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>},\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>})+\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}(\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>},\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}).}$$

This ████ ████████ is ██████████ a ███████████ of the ████ ███████████ ██████–███████ ██████████, which ███████ $${\displaystyle \left|⟨\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>},\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}⟩\right|\le \Vert \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}\Vert \Vert \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}\Vert }$$ with ████████ if and ████ if ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}$ and ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}$ are ████████ █████████.

With a ████████ ████████ ███████ in this ███, ███ █████ ███████ █████ is a ██████ █████. An █████ ███████ █████ is █████████ █████ as a ███-███████ █████. ███ ███-███████ █████ that is ████████████ ████ a ████████ █████ is a ███████ █████.

The ████████████ of █ is █████████ using a ████ of the ██████ █████████ for █████████ in █: a ███-███████ █████ █ is ████████ if █████ ██████ ████████ █████████ with ███████ to this ████ to an ███████ in the █████. ████████████ can be █████████████ by the following ██████████ █████████: if a ██████ of ███████ $${\displaystyle \sum _{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}=\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}^{\mathrm{\infty }}{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}_{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}}$$ █████████ ██████████ in the █████ that ████ the ██████ █████████ in █, in the █████ that the ███████ ████ ████████ to an ███████ of █.

As a ████████ ██████ █████, ███████ ██████ are by ██████████ ████ ██████ ██████. As ████ they are ███████████ ██████ ██████, in which ███████████ ███████ like the ████████ and ██████████ of ███████ are ████ ███████. Of ███████ ██████████ is the ██████ of a ██████ ██████ ████████ of a ███████ █████ that, with the █████ ███████ ███████ by ███████████, is ████ ████████ (being a ██████ ███ in a ████████ ██████ █████) and █████████ a ███████ █████ in ███ ███ █████.

██████ ███████: ████████ ██████

The ████████ █████ ${\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}^{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}$ ████████ of ███ ████████ █████████ █ = (█_█, █_█, ...) of ███████ ███████ ████ that the ██████ $${\displaystyle \sum _{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}=\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}^{\mathrm{\infty }}|{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}_{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}{|}^{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}}$$ of ███ ███████ █████ █████████. The █████ ███████ on ${\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}^{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}$ is ███████ by $${\displaystyle ⟨\mathbf{<span\; word-length="1"\; class="word\; redacted">█</span>},\mathbf{<span\; word-length="1"\; class="word\; redacted">█</span>}⟩=\sum _{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}=\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}^{\mathrm{\infty }}{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}_{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}{\overline{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}}_{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}.}$$ The ██████ for the █████ ███████ █████████ as a ███████████ of the ██████–███████ ██████████ and the ███████ ███████████ of the ███ ██████ of ███████ █████.

████████████ of the █████ █████ ████████ that ████████ a ██████ of ████████ from ${\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}^{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}$ █████████ ██████████ (in ████), ████ it █████████ to an ███████ of ${\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}^{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}$. The █████ is █████ in ████████████ ████████, and ███████ ████████████ ██████ of ████████ of the █████ to be ███████████ with the ████ ████ as ██████ of ███████ ███████ (or ███████ in a ██████-███████████ █████████ █████).