힐베르트 공간에서의 베셀 시퀀스
📂힐베르트공간 힐베르트 공간에서의 베셀 시퀀스 정의 힐베르트 공간 H H H 시퀀스 { v k } k ∈ N ⊂ H \left\{ \mathbf{v}_{k} \right\}_{k \in \mathbb{N}} \subset H { v k } k ∈ N ⊂ H B > 0 B > 0 B > 0 { v k } k ∈ N \left\{ \mathbf{v}_{k} \right\}_{k \in \mathbb{N}} { v k } k ∈ N 베셀 시퀀스 bessel sequence 라 하고 B B B 베셀 바운드 bessel bound 라 한다.
∑ k = 1 ∞ ∣ ⟨ v , v k ⟩ ∣ 2 ≤ B ∥ v ∥ 2 , ∀ v ∈ H
\sum_{k=1}^{\infty} \left| \left\langle \mathbf{v} , \mathbf{v}_{k} \right\rangle \right|^{2 } \le B \left\| \mathbf{v} \right\|^{2}, \quad \forall \mathbf{v} \in H
k = 1 ∑ ∞ ∣ ⟨ v , v k ⟩ ∣ 2 ≤ B ∥ v ∥ 2 , ∀ v ∈ H
설명 베셀 시퀀스는 직관적으로 봤을 때 무한차원 벡터 v \mathbf{v} v
정리 힐베르트 공간 H H H 시퀀스 { v k } k ∈ N ⊂ H \left\{ \mathbf{v}_{k} \right\}_{k \in \mathbb{N}} \subset H { v k } k ∈ N ⊂ H B > 0 B > 0 B > 0
{ v k } k ∈ N \left\{ \mathbf{v}_{k} \right\}_{k \in \mathbb{N}} { v k } k ∈ N B B B
다음과 같이 정의된 작용소 T T T ∥ T ∥ ≤ B \left\| T \right\| \le \sqrt{B} ∥ T ∥ ≤ B
T : l 2 → H T { c k } k ∈ N : = ∑ k ∈ N c k v k
T : l^{2} \to H
\\ T \left\{ c_{k} \right\}_{k \in \mathbb{N}} := \sum_{k \in \mathbb{N}} c_{k} \mathbf{v}_{k}
T : l 2 → H T { c k } k ∈ N := k ∈ N ∑ c k v k
증명 ( ⟹ ) (\implies) ( ⟹ )
{ v k } k ∈ N \left\{ \mathbf{v}_{k} \right\}_{k \in \mathbb{N}} { v k } k ∈ N B B B { c k } k ∈ N ∈ l 2 \left\{ c_{k} \right\}_{k \in \mathbb{N}} \in l^{2} { c k } k ∈ N ∈ l 2 T { c k } k ∈ N = ∑ k ∈ N c k v k T \left\{ c_{k} \right\}_{k \in \mathbb{N}} = \sum_{k \in \mathbb{N}} c_{k} \mathbf{v}_{k} T { c k } k ∈ N = ∑ k ∈ N c k v k 자연수 n > m n > m n > m
∥ ∑ k = 1 n c k v k − ∑ k = 1 m c k v k ∥ = ∥ ∑ k = m + 1 n c k v k ∥ = sup ∥ w ∥ = 1 ∣ ∑ k = m + 1 n ⟨ c k v k , w ⟩ ∣
\left\| \sum_{k =1}^{n} c_{k} \mathbf{v}_{k} - \sum_{k =1}^{m} c_{k} \mathbf{v}_{k} \right\| = \left\| \sum_{k = m + 1}^{n} c_{k} \mathbf{v}_{k} \right\| = \sup_{\left\| \mathbf{w} \right\| = 1 } \left| \sum_{k = m + 1}^{n} \left\langle c_{k} \mathbf{v}_{k} , \mathbf{w} \right\rangle \right|
k = 1 ∑ n c k v k − k = 1 ∑ m c k v k = k = m + 1 ∑ n c k v k = ∥ w ∥ = 1 sup k = m + 1 ∑ n ⟨ c k v k , w ⟩
삼각 부등식 에 의해
sup ∥ w ∥ = 1 ∣ ∑ k = m + 1 n ⟨ c k v k , w ⟩ ∣ ≤ sup ∥ w ∥ = 1 ∑ k = m + 1 n ∣ ⟨ c k v k , w ⟩ ∣
\sup_{\left\| \mathbf{w} \right\| = 1 } \left| \sum_{k = m + 1}^{n} \left\langle c_{k} \mathbf{v}_{k} , \mathbf{w} \right\rangle \right| \le \sup_{\left\| \mathbf{w} \right\| = 1 } \sum_{k = m + 1}^{n} \left| \left\langle c_{k} \mathbf{v}_{k} , \mathbf{w} \right\rangle \right|
∥ w ∥ = 1 sup k = m + 1 ∑ n ⟨ c k v k , w ⟩ ≤ ∥ w ∥ = 1 sup k = m + 1 ∑ n ∣ ⟨ c k v k , w ⟩ ∣
코시-슈바르츠 부등식 에 의해
sup ∥ w ∥ = 1 ∑ k = m + 1 n ∣ ⟨ c k v k , w ⟩ ∣ ≤ ( ∑ k = m + 1 n ∣ c k ∣ 2 ) 1 / 2 sup ∥ w ∥ = 1 ( ∑ k = m + 1 n ∣ ⟨ v k , w ⟩ ∣ 2 ) 1 / 2 ≤ ( ∑ k = m + 1 n ∣ c k ∣ 2 ) 1 / 2 B
\begin{align*}
& \sup_{\left\| \mathbf{w} \right\| = 1 } \sum_{k = m + 1}^{n} \left| \left\langle c_{k} \mathbf{v}_{k} , \mathbf{w} \right\rangle \right|
\\ \le & \left( \sum_{k = m + 1}^{n} \left| c_{k} \right|^{2} \right)^{1/2} \sup_{\left\| \mathbf{w} \right\| = 1 } \left( \sum_{k = m + 1}^{n} \left| \left\langle \mathbf{v}_{k} , \mathbf{w} \right\rangle \right|^{2} \right)^{1/2}
\\ \le & \left( \sum_{k = m + 1}^{n} \left| c_{k} \right|^{2} \right)^{1/2} \sqrt{B}
\end{align*}
≤ ≤ ∥ w ∥ = 1 sup k = m + 1 ∑ n ∣ ⟨ c k v k , w ⟩ ∣ ( k = m + 1 ∑ n ∣ c k ∣ 2 ) 1/2 ∥ w ∥ = 1 sup ( k = m + 1 ∑ n ∣ ⟨ v k , w ⟩ ∣ 2 ) 1/2 ( k = m + 1 ∑ n ∣ c k ∣ 2 ) 1/2 B
{ c k } k ∈ N ∈ l 2 \left\{ c_{k} \right\}_{k \in \mathbb{N}} \in l^{2} { c k } k ∈ N ∈ l 2 { ∑ k = 1 n c k v } n = 1 ∞ ⊂ H \displaystyle \left\{ \sum_{k=1}^{n} c_{k} \mathbf{v} \right\}_{n=1}^{\infty} \subset H { k = 1 ∑ n c k v } n = 1 ∞ ⊂ H T T T T T T
∥ T { c K } k ∈ N ∥ = sup ∥ w ∥ = 1 ∣ ⟨ T { c k } k ∈ N , w ⟩ ∣ ≤ B ( ∑ k ∈ N ∣ c k ∣ 2 ) 1 / 2 = B ∥ { c k ∈ N } ∥ 2
\begin{align*}
\left\| T \left\{ c_{K} \right\}_{k \in \mathbb{N}} \right\| =& \sup_{\left\| \mathbf{w} \right\| = 1 } \left| \left\langle T \left\{ c_{k} \right\}_{k \in \mathbb{N}} , \mathbf{w} \right\rangle \right|
\\ \le & \sqrt{B} \left( \sum_{k \in \mathbb{N}} \left| c_{k} \right|^{2} \right)^{1/2}
\\ =& \sqrt{B} \left\| \left\{ c_{k \in \mathbb{N}} \right\} \right\|_{2}
\end{align*}
T { c K } k ∈ N = ≤ = ∥ w ∥ = 1 sup ⟨ T { c k } k ∈ N , w ⟩ B ( k ∈ N ∑ ∣ c k ∣ 2 ) 1/2 B ∥ { c k ∈ N } ∥ 2
이므로 양변을 ∥ { c k ∈ N } ∥ 2 \left\| \left\{ c_{k \in \mathbb{N}} \right\} \right\|_{2} ∥ { c k ∈ N } ∥ 2 ∥ T ∥ ≤ B \left\| T \right\| \le \sqrt{B} ∥ T ∥ ≤ B
( ⟸ ) (\impliedby) ( ⟸ )
{ v k } k ∈ N \left\{ \mathbf{v}_k \right\}_{k \in \mathbb{N}} { v k } k ∈ N H H H T : l 2 → H T : l^{2} \to H T : l 2 → H
T { c k } k ∈ N : = ∑ k = 1 ∞ c k v k
T \left\{ c_{k} \right\}_{k \in \mathbb{N}} := \sum_{k=1}^{\infty} c_{k} \mathbf{v}_{k}
T { c k } k ∈ N := k = 1 ∑ ∞ c k v k
그러면 T T T T ∗ : H → l 2 T^{ \ast } : H \to l^{2} T ∗ : H → l 2
T ∗ v = { ⟨ v , v k ⟩ H } k ∈ N
T^{ \ast } \mathbf{v} = \left\{ \left\langle \mathbf{v} , \mathbf{v}_{k} \right\rangle_{H} \right\}_{k \in \mathbb{N}}
T ∗ v = { ⟨ v , v k ⟩ H } k ∈ N
그 뿐만 아니라, 모든 v ∈ H \mathbf{v} \in H v ∈ H
∑ k = 1 ∞ ∣ ⟨ v , v k ⟩ H ∣ 2 ≤ ∥ T ∥ 2 ∥ v ∥ H 2
\sum_{k=1}^{\infty} \left| \left\langle \mathbf{v} , \mathbf{v}_{k} \right\rangle_{H} \right|^{2} \le \left\| T \right\|^{2} \left\| \mathbf{v} \right\|_{H}^{2}
k = 1 ∑ ∞ ∣ ⟨ v , v k ⟩ H ∣ 2 ≤ ∥ T ∥ 2 ∥ v ∥ H 2
T T T ∥ T ∥ ≤ B \left\| T \right\| \le \sqrt{B} ∥ T ∥ ≤ B
∑ k = 1 ∞ ∣ ⟨ v , v k ⟩ ∣ 2 ≤ ∥ T ∥ 2 ∥ v ∥ 2 ≤ B ∥ v ∥ 2
\sum_{k=1}^{\infty} \left| \left\langle \mathbf{v} , \mathbf{v}_{k} \right\rangle \right|^{2} \le \left\| T \right\|^{2} \left\| \mathbf{v} \right\|^{2} \le B \left\| \mathbf{v} \right\|^{2}
k = 1 ∑ ∞ ∣ ⟨ v , v k ⟩ ∣ 2 ≤ ∥ T ∥ 2 ∥ v ∥ 2 ≤ B ∥ v ∥ 2
따라서 { v k } k ∈ N \left\{ \mathbf{v}_k \right\}_{k \in \mathbb{N}} { v k } k ∈ N B B B
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