████ █████

████ ██████: ████ ██████ ████████

███ ███████ ██████ █████ █████ ██████ ██ ████ ██████ ████████ ██████ ██ ██████ ██████ ███████، ██████ ██████ ██████، █████ ████ ██████ █^█. █████ ██████ ███████ ████ ██████ █ ██ █████ ████ ███ █████ █ ⋅ █. ██ █████ █ ██ ███████████ ██████████، ███ ████ ██████ ███████ ████:$${\displaystyle \left(\begin{array}{c}{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}_{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}\\ {\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}_{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}\\ {\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}_{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}\end{array}\right)\cdot \left(\begin{array}{c}{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}_{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}\\ {\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}_{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}\\ {\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}_{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}\end{array}\right)={\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}_{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}_{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}+{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}_{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}_{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}+{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}_{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}_{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}.}$$

██████ ██████ ██████ ███████ ███████:

• ██████ ██ █ █ █: █ ⋅ █ = █ ⋅ █
• ███ ████ ███ ██ ██████: (██_█ + ██_█) ⋅ █ = ██_█ ⋅ █ + ██_█ ⋅ █ ███ ████ █████ █ █ █ █████████ █_█ ██_█ ██
• ████ ███████: █████ ████████ █، █ ⋅ █ ≥ █ ، ██████ █████ ███ ████ ███ ███ █ = █

██████ ██████ ████ ███ ██████ ███ █████ ████████ █████████ ████ ███████ ██████، ████ ███████ ███████ (███████). ████ ██████ ██████ ████ ██████ ███████ ███████ ████ ██ ████ ████ █████ (█████). ██ ████ ████ █████ ██ █████ ██████ ██ ████ ████ ██████. ███████ ████████ ██████ ██████ ████ █████ ████████ █████████ ██ ███ █████ ████ (██ █████) ██████، ██████ ████ ██ ||█|| ، ████████ █ ███ ██████ █ █ █ ██ ████ ██████:$${\displaystyle \mathbf{<span\; word-length="1"\; class="word\; redacted">█</span>}\cdot \mathbf{<span\; word-length="1"\; class="word\; redacted">█</span>}=\Vert \mathbf{<span\; word-length="1"\; class="word\; redacted">█</span>}\Vert \Vert \mathbf{<span\; word-length="1"\; class="word\; redacted">█</span>}\Vert \mathrm{<span\; word-length="3"\; class="word\; redacted">███</span>}\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}.}$$

████ ███████ ████████ █████ █████████ ██ ██████ ████████ █████ ███ ████ ████████، ███████ ███ ██████ █████ ████████ ████ ████████. ███ █████████ ████████$${\displaystyle \sum _{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}=\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}^{\mathrm{\infty }}{\mathbf{<span\; word-length="1"\; class="word\; redacted">█</span>}}_{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}}$$███████ ██ ██████ ██ █^█ ██████ ████ ████ █████ ██ ██████ █████ ███████ ███ ███████ █████ ██ ███████ ████████:█████ ███ ██ █████ ██ ███████ ██ █████، ███ ███████ ██ ████████ ████ ██████ █████ ██████ ████ ██ ████ █████ █ ██ ██████ ████████، █████ ██$${\displaystyle \Vert \mathbf{<span\; word-length="1"\; class="word\; redacted">█</span>}-\sum _{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}=\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}^{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}{\mathbf{<span\; word-length="1"\; class="word\; redacted">█</span>}}_{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}\Vert \to \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}\text{<span word-length="2" class="word redacted">██</span> }\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}\to \mathrm{\infty }.}$$████ ███ ███████ ██ ██████ ██████ ████████: ██ ██ █████████ ████ ██████ █████ ██████ ████ ███████ ███████.

█████ ██ ████ ███████ ███████ ███████ ██ ██████ █████. ████████ ██████ ██████ ██ ██ █ ████ █████ █████، ███████ ███████ |█| ██████ ████ █████ ████████ █████ █ ██ ██████ ██████: $${\displaystyle |\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}{|}^{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}=\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}\overline{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}.}$$ ███ ███ █ = █ + ██ ██ █████ █ ███ █████ ███████ ████████، ███ ███████ ██ █████ ████████ █████ ███████ ███████:$${\displaystyle |\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}|=\sqrt{{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}^{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}+{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}^{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}}.}$$██████ ███████ ████ ██ ███████ ███████ █ █ █ ██ █ ██ █████ █:$${\displaystyle ⟨\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>},\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}⟩=\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}\overline{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}.}$$███████ ████ ██████. █████ ███████ ██ ⟨█, █⟩ ████ ██████ ██████ ████████ ███████ █████ ███████.

██████ ██████ ██ ██████ █^█ ████ █████ ██████ ██ █████ ██ ███████ ███████ █ = (█_█, █_█). █████ ██████ ███████ ██ █ ██ ████ ███ █ = (█_█, █_█)████ ███████ ███████:$${\displaystyle ⟨\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>},\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}⟩={\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}_{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}\overline{{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}_{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}}+{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}_{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}\overline{{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}_{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}}.}$$█████ ███████ ██ ⟨█, █⟩██ ██████ ██████ ████████ █████ ███████. ███ ██████ ███████ █████ ████ ██████ ████████ "█████████ █████████"، ███ ████ ██ █████ ███████ ███ █ █ █ ██ █████ ████:$${\displaystyle ⟨\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>},\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}⟩=\overline{⟨\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>},\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}⟩}.}$$

█████

████ █████ █ ██ ████ ████ █████ █████ ██ ████ ███ ████ ████ ████ ████ ██ ████ ███████ ███████ ██ ██████ ███████.

█████ ████ ██ █ ██ ████ ████ █████ ████ █████ ████ ██ █ ██ ████ █████ ████ ████ ████ █████ ${\displaystyle ⟨\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>},\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}⟩}$ ████ ██ ███ ██ ███████ ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>},\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}$ ██ █ ████ ████، ██████ ███ ██████ ████████ ███████:

1. ██████ ███████ ██████ ██████؛ ██ ██ ██████ ███████ ████ ██ ███████ █████ ███████ ██████ ██████ ███████ ████ ███████ ██ █████ ███████:${\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>}⟩}.}$$${\displaystyle ⟨\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>},\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}⟩=\overline{⟨\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>},\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}⟩}.}$$
2. ██████ ███████ ███ ████ ███ ██ ██████ █████ ███████ ███████ ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}$ █ ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>},}$ $${\displaystyle ⟨\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}_{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}+\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}_{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}},\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}⟩=\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}⟨{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}_{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}},\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}⟩+\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}⟨{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}_{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}},\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}⟩.}$$
3. ██████ ███████ █████ ██ ████ ██ ██████ █████:

██████ ██ █████████ █ █ █ ██ ██████ ███████ ██████ ██ ██████ ███ ███ "██████████ ███"، █████ ████ "█████████ ██████"، █████ ██████ ██ ██████، █████ ██:$${\displaystyle ⟨\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>},\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}_{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}+\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}_{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}⟩=\overline{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}⟨\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>},{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}_{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}⟩+\overline{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}⟨\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>},{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}_{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}⟩.}$$████ ██████ ███████ ███████ ████ ████ ███████، ███ ██ █ ██ ████ █████ █████ ███ ██████ ███████ ████ ████ ██████. ███ ███ ██████ ███████ █████ ███████ █████ ██████ █${\displaystyle (\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>},\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>},⟨\cdot ,\cdot ⟩)}$ █████ ████ █████.

███████ ██ ████ ██████ ██████ ████ ███ █████ ██████: $${\displaystyle \Vert \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}\Vert =\sqrt{⟨\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>},\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}⟩},}$$

████████ ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}$ ███ ██████ ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>},\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}$ ██ █ ████ ██ ████ ███████ ██████ ██████:$${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}(\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>},\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>})=\Vert \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}-\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}\Vert =\sqrt{⟨\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}-\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>},\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}-\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}⟩}.}$$█████ ███████ ███ ████ ████ ████ ███████ ██ ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}$ █ ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}$ (███ ██████ ████ ██ ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}$ ███ ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}$ ██ ██ ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}$ ███ ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}$) ██████ ██ ███████ ███ ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}$ ██████ ███، ████ ███ ████████ ███ ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}$ █ ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}$ ███ ██ ████ █████، ██████ ███ ███████ ██████ █████، ███ ████ ██ ███ ███ █████ ██████ ███ ██ ████ ██ ██████ █████ ████ ███████ ███████:$${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}(\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>},\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>})\le \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}(\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>},\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>})+\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}(\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>},\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}).}$$

███ ███████ ███████ ██ ██ ██████ ████ ████████ ████ -██████، █████ ████ ██:$${\displaystyle \left|⟨\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>},\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}⟩\right|\le \Vert \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}\Vert \Vert \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}\Vert }$$████████ ███ ████ ███ ███ ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}$ █ ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}$ ███ ███████ ████.

█████ ████ █████ █████ ████ █████، ███ ██ ████ ████ █████ ██ ████ ████، █████ ██████ ████ «████ ████████ ██ ███ █████» (█████████ ███-███████ █████). ██ ████ ██ ███ █████ █████ ████ ████ ███ ████ ██████.

████ ██ ██████ █ ████████ █████ ██ ██████ ████ ██████████ ██ █: █████ ██ ███ █████ █ █████ ███ ███ ██ ███████ █████ ██████ ███████ ███████ ███ ████ ██ ██████. ████ ███ ████████ ██████ ███████ ██████: ███ ████ ███████ ████████$${\displaystyle \sum _{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}=\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}^{\mathrm{\infty }}{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}_{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}}$$██████ ████ ████ █████ █████ █████████ ██████ ██ █، █████ ██ ████████ ███████ ██████ ███ ████ ██ █

████ ██████ █████ ███████ █████، █████ ████████ ██ ████ ██████ █████. ████ ███ ██████ ██████ █████████، ███ ██████ ███████████ █████ ████ ███ ███ ██████ ███████ █████████ ███████. █████ ██████ ██████ █████ ██████ ██ ████ █████ █████ ████، ███ ████ █████ ██████ ████ ██ ██████ ███████ ██████، (████ ██████ █████ ██ ████ ████ ████) ████████ ███ ████ ██████ ██ ██ ████.

██████ ██████: ██████ ██████████

████ █████████ ████ ████ ██ ██ █^█ █████ ██ ████ ██████████ ██████████ █ = (█_█, █_█, …) ██ ███████ ███████ ████ ████ █████████$${\displaystyle \sum _{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}=\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}^{\mathrm{\infty }}|{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}_{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}{|}^{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}}$$███████. █████ ██████ ███████ ██ █^█ ███ █████ ██████:$${\displaystyle ⟨\mathbf{<span\; word-length="1"\; class="word\; redacted">█</span>},\mathbf{<span\; word-length="1"\; class="word\; redacted">█</span>}⟩=\sum _{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}=\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}^{\mathrm{\infty }}{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}_{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}\overline{{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}_{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}},}$$███████ ████████ ████ -█████، ███ █████████ ███████ ██████.

███ ██████ ██████ █████ █████ ██ ███████ ██ ███████ ██ █^█ ██████ █████ ███████ ███████، ███ █████ ██████ ██ ████ ██ █^█. █████ ███████ ██████ ██ ███████ ███████، █████ ████████ ██ ███████ ██ █████ ██████ ████ ███████ ███ ██ ████ ███████ █████ █████ (██ ██████ ██ ████ ██████ █████ ███████).