Definition (l 2 ( Z + ; R ) l_{2}(\mathbb{Z}_{+};\mathbb{R}) l 2 ( Z + ; R )
l 2 ( Z + ; R ) : = { x ∈ Γ ( Z + ; R ) : ∥ x ∥ 2 = ( ∑ i ∈ Z + ∣ x ( i ) ∣ 2 ) 1 / 2 < ∞ } l_{2}(\mathbb{Z}_{+};\mathbb{R}):=\left\{ x\in\Gamma(\mathbb{Z}_{+};\mathbb{R}):\lVert x \rVert _{2}=\left( \sum_{i\in \mathbb{Z}_{+}}\left| x(i) \right| ^{2} \right)^{1/2}<\infty \right\} l 2 ( Z + ; R ) := ⎩ ⎨ ⎧ x ∈ Γ ( Z + ; R ) : ∥ x ∥ 2 = i ∈ Z + ∑ ∣ x ( i ) ∣ 2 1/2 < ∞ ⎭ ⎬ ⎫
Theorem (2.3.6)
The Hilbert space l 2 ( Z + ; R ) l_{2}(\mathbb{Z}_{+};\mathbb{R}) l 2 ( Z + ; R ) inner product ⟨ h 1 , h 2 ⟩ = ∑ n ∈ Z + h 1 ( n ) h 2 ( n ) , \langle h_{1}, h_{2} \rangle =\sum_{n\in \mathbb{Z}_{+}}h_{1}(n)h_{2}(n), ⟨ h 1 , h 2 ⟩ = n ∈ Z + ∑ h 1 ( n ) h 2 ( n ) , separable .
Definition (L 2 L^{2} L 2
L 2 ( R + ; R ) = { f ∈ Γ ( R + ; R ) : ∥ f ∥ 2 = ( ∫ R + ∣ f ( x ) ∣ 2 λ ( d x ) ) 1 / 2 } L^{2}(\mathbb{R}_{+};\mathbb{R})= \left\{ f\in\Gamma(\mathbb{R}_{+};\mathbb{R}):\lVert f \rVert _{2}=\left( \int\limits _{\mathbb{R}_{+}}|f(x)|^{2} \, \lambda(dx) \right)^{1/2} \right\} L 2 ( R + ; R ) = ⎩ ⎨ ⎧ f ∈ Γ ( R + ; R ) : ∥ f ∥ 2 = R + ∫ ∣ f ( x ) ∣ 2 λ ( d x ) 1/2 ⎭ ⎬ ⎫
Theorem (2.3.7)
Any function in C ( [ 0 , 1 ] ; R ) C([0,1];\mathbb{R}) C ([ 0 , 1 ] ; R ) polynomial under the supremum norm .
Theorem (2.3.8)
The set C ( [ 0 , 1 ] ; R ) C([0,1];\mathbb{R}) C ([ 0 , 1 ] ; R ) L 2 ( [ 0 , 1 ] ; R ) L^{2}([0,1];\mathbb{R}) L 2 ([ 0 , 1 ] ; R )
Theorem (2.3.9)
The space L 2 ( [ 0 , 1 ] , R ) L^{2}([0,1],\mathbb{R}) L 2 ([ 0 , 1 ] , R ) separable .
Theorem (2.3.10)
L 2 ( R + ; R ) L^{2}(\mathbb{R}_{+};\mathbb{R}) L 2 ( R + ; R ) separable .
Theorem (2.3.11)
L 2 ( [ 1 , ∞ ) ; R ) L^{2}([1,\infty);\mathbb{R}) L 2 ([ 1 , ∞ ) ; R ) dense in L 1 ( [ 1 , ∞ ) ; R ) L^{1}([1,\infty);\mathbb{R}) L 1 ([ 1 , ∞ ) ; R )
Theorem (2.3.12)
C c C_{c} C c dense in L 1 ( R ; R ) L^{1}(\mathbb{R};\mathbb{R}) L 1 ( R ; R )