Возьмём в пространстве
L
2
(
a
,
b
)
{\displaystyle L_{2}\left(a,b\right)}
{
φ
i
(
t
)
}
{\displaystyle \left\{\varphi _{i}(t)\right\}}
x
(
t
)
∈
L
2
(
a
,
b
)
{\displaystyle x(t)\in L_{2}\left(a,b\right)}
x
(
t
)
=
∑
i
=
1
∞
ξ
i
φ
i
(
t
)
,
ξ
i
=
(
x
,
φ
i
)
{\displaystyle x(t)=\sum _{i=1}^{\infty }\xi _{i}\varphi _{i}(t),\xi _{i}=\left(x,\varphi _{i}\right)}
равенства Парсеваля
∑
i
=
1
∞
ξ
i
2
=
‖
x
‖
2
<
∞
{\displaystyle \sum _{i=1}^{\infty }\xi _{i}^{2}=\left\|x\right\|^{2}<\infty }
коэффициентов Фурье функции
x
(
t
)
∈
L
2
(
a
,
b
)
{\displaystyle x(t)\in L_{2}\left(a,b\right)}
x
∈
l
2
{\displaystyle x\in l_{2}}
l
2
{\displaystyle l_{2}}
x
=
{
ξ
1
,
ξ
2
,
.
.
.
,
ξ
n
,
.
.
.
}
{\displaystyle x=\left\{\xi _{1},\xi _{2},...,\xi _{n},...\right\}}
x
(
t
)
→
x
{\displaystyle x(t)\rightarrow x}
y
¯
=
{
η
1
,
η
2
,
.
.
.
,
η
n
,
.
.
.
}
{\displaystyle {\bar {y}}=\left\{\eta _{1},\eta _{2},...,\eta _{n},...\right\}}
l
2
{\displaystyle l_{2}}
L
2
(
a
,
b
)
{\displaystyle L_{2}\left(a,b\right)}
∑
i
=
1
∞
η
i
φ
i
(
t
)
{\displaystyle \sum _{i=1}^{\infty }\eta _{i}\varphi _{i}(t)}
{
φ
i
(
t
)
}
{\displaystyle \left\{\varphi _{i}(t)\right\}}
s
n
(
t
)
=
∑
i
=
1
n
η
i
φ
i
(
t
)
{\displaystyle s_{n}(t)=\sum _{i=1}^{n}\eta _{i}\varphi _{i}(t)}
‖
s
n
+
p
−
s
n
‖
2
=
‖
∑
i
=
n
+
1
n
+
p
η
i
φ
i
(
t
)
‖
2
=
∑
i
=
n
+
1
n
+
p
η
i
2
→
0
{\displaystyle \left\|s_{n+p}-s_{n}\right\|^{2}=\left\|\sum _{i=n+1}^{n+p}\eta _{i}\varphi _{i}(t)\right\|^{2}=\sum _{i=n+1}^{n+p}\eta _{i}^{2}\rightarrow 0}
n
→
∞
{\displaystyle n\rightarrow \infty }
p
>
0
{\displaystyle p>0}
∑
i
=
1
∞
η
i
2
{\displaystyle \sum _{i=1}^{\infty }\eta _{i}^{2}}
L
2
(
a
,
b
)
{\displaystyle L_{2}\left(a,b\right)}
∑
i
=
1
∞
η
i
φ
i
(
t
)
{\displaystyle \sum _{i=1}^{\infty }\eta _{i}\varphi _{i}(t)}
η
i
{\displaystyle \eta _{i}}
y
(
t
)
∈
L
2
(
a
,
b
)
{\displaystyle y(t)\in L_{2}\left(a,b\right)}
y
¯
{\displaystyle {\bar {y}}}
y
¯
→
y
(
t
)
{\displaystyle {\bar {y}}\rightarrow y(t)}
L
2
(
a
,
b
)
{\displaystyle L_{2}\left(a,b\right)}
l
2
{\displaystyle l_{2}}
x
(
t
)
+
y
(
t
)
=
∑
i
=
1
∞
ξ
i
φ
i
(
t
)
+
∑
i
=
1
∞
η
i
φ
i
(
t
)
=
∑
i
=
1
∞
(
ξ
i
+
η
i
)
φ
i
(
t
)
{\displaystyle x(t)+y(t)=\sum _{i=1}^{\infty }\xi _{i}\varphi _{i}(t)+\sum _{i=1}^{\infty }\eta _{i}\varphi _{i}(t)=\sum _{i=1}^{\infty }\left(\xi _{i}+\eta _{i}\right)\varphi _{i}(t)}
λ
x
(
t
)
=
λ
∑
i
=
1
∞
ξ
i
φ
i
(
t
)
=
∑
i
=
1
∞
λ
ξ
i
φ
i
(
t
)
{\displaystyle \lambda x(t)=\lambda \sum _{i=1}^{\infty }\xi _{i}\varphi _{i}(t)=\sum _{i=1}^{\infty }\lambda \xi _{i}\varphi _{i}(t)}
x
(
t
)
↔
x
,
y
(
t
)
↔
y
{\displaystyle x(t)\leftrightarrow x,y(t)\leftrightarrow y}
x
(
t
)
+
y
(
t
)
↔
x
+
y
,
λ
x
(
t
)
↔
λ
x
{\displaystyle x(t)+y(t)\leftrightarrow x+y,\lambda x(t)\leftrightarrow \lambda x}
x
(
t
)
,
y
(
t
)
∈
L
2
(
a
,
b
)
{\displaystyle x(t),y(t)\in L_{2}\left(a,b\right)}
‖
x
(
t
)
−
y
(
t
)
‖
2
=
‖
∑
i
=
1
∞
(
ξ
i
−
η
i
)
φ
i
(
t
)
‖
2
=
∑
i
=
1
∞
(
ξ
i
−
η
i
)
2
=
‖
x
−
y
‖
2
{\displaystyle \left\|x(t)-y(t)\right\|^{2}=\left\|\sum _{i=1}^{\infty }\left(\xi _{i}-\eta _{i}\right)\varphi _{i}(t)\right\|^{2}=\sum _{i=1}^{\infty }\left(\xi _{i}-\eta _{i}\right)^{2}=\left\|x-y\right\|^{2}}
L
2
(
a
,
b
)
{\displaystyle L_{2}\left(a,b\right)}
l
2
{\displaystyle l_{2}}
