Hilbert Space
Banach Space
Before we define Hilbert space, it is helpful to define normed linear spaces and Banach Spaces.
A vector space \(X\) is called normed linear space if for each element \(x \in X \) there is a number \( \|x\| \) called norm such that following conditions are satisfied
- 1. \( \|x\| \geq 0 \) with \( \|x\|=0\) if and only if \(x=0\)
- 2. \(\|cx\|=|c|\|x\|\) for a scalar \(c\)
- 3. \( \|x+y\| \leq \|x\|+\|y\| \)
Convergence and Completeness condition: Let \(X\) be a normed linear space
- 1. A sequence of vectors \( {x_{n}}\) converges to \( x \in X\) if \(\lim_{n\rightarrow \infty} \|x-x_{n}\|=0\). Putting it in a different way, if \( \forall \epsilon \gt 0 \), \(\exists N \gt 0\),\( \forall n \geq N \), \( \|x-x_{n}\| \gt \epsilon\)
- 2. Cauchy Sequence: A vector sequence \( {x_{n}}\) is Cauchy if \(\lim_{m,n\rightarrow \infty} \|x_{m}-x_{n}\|=0\). Putting it in a different way, if \( \forall \epsilon \gt 0 \), \(\exists N \gt 0\),\( \forall m,n \geq N \), \( \|x_{m}-x_{n}\| \gt \epsilon\)
- 3. We say that \(X\) is complete if every Cauchy sequence in \(X\) is a convergent sequence. A complete normed linear space is called a BANACH SPACE.
Hilbert Space
A vector space \(H\) is called a Hilbert space if for each pair \((x,y)\) of elements in the space \(H\) there is a unique number called inner product, denoted by \( \lt x,y \gt \) ,subject to following three conditions
- 1. Linearity : \[ \lt \alpha x + \beta y ,z \gt = \alpha \lt x,z \gt + \beta \lt y,z \gt \] where \(x,y,z \in H \) and \(\alpha, \beta \in C\)
- 2. Conjugated linearity: \( \lt x,y \gt = \overline{\lt y,x \gt} \)
- 3. \((\lt x,x \gt) \gt 0\) for \( x \neq 0 \)
Properties of Hilbert Space
- 1. Norm: For an element \(x \in H \) , the norm is given by \[ \|x\|=\sqrt{\lt x,x \gt} \]
- 2. Cauchy-Schwartz Inequality holds for any elements \(x,y \in H\) \[ |\lt x,y \gt| \leq \|x\|\|y\| \]
- 3. Orthogonality: Vectors \(x\) and \(y\) are said to be orthogonal if \(\lt x,y \gt = 0\). Orthogonality is written as \(x \bot y \) Euclidean Pythagoran theorem holds in Hilbert Space \[ x \bot y \Rightarrow \|x+y\|^{2}=\|x\|^{2}+\|y\|^{2} \]
Operators on Hilbert Spaces
- Adjoint Operator: A linear operator \(U^{*}:H_{2} \rightarrow H_{1}\) is said to be adjoint of the operator \(U:H_{1} \rightarrow H_{2}\) when \[ \lt Ux,y \gt_{H_{2}}=\lt x,U^{*}y \gt_{H_{1}} \] for every \(x \in H_{1}\) and every \(y \in H_{2}\)
- Self-Adjoint Operator: If \(U=U^{*}\) then \(U\) is called self-adjoint operator.
- Unitary Operator: \( UU^{*} =U^{*}U=I\)