A distribution on a manifold (M) is a linear functional on the functions on (M).
A distribution is a linear functional on of infinitely differentiable functions on with compact supports such that for every compact set there exist constants and satisfying (2.1).
Suppose that L is a linear functional applied to the arbitrary function h ( x ), and then consider the following special function: H ( x ; h, λ ) = h ( x ) + λ L ( h ) ( λ ∈ R ∖ { 0 } ).
As a distribution, the Dirac delta is a linear functional on the space of test functions and is defined by \delta[\varphi] = \varphi(0)\, for every test function φ.
Recall that a distribution u is a linear functional on C c ∞ ( R ) of infinitely differentiable functions on ℝ with compact supports such that for every compact set K ⊂ R, there exist constants C > 0 and N ∈ N 0 satisfying | 〈 u, φ 〉 | ≤ C ∑ | α | ≤ N sup | ∂ α φ |. for all φ ∈ C c ∞ ( R ) with supports contained in K.
Recall that a distribution u is a linear functional on C c ∞ ( ℝ m ) of infinitely differentiable functions on ℝ m with compact supports such that for every compact set K ⊂ ℝ m there exist constants C > 0 and N ∈ ℕ0 satisfying ∣ 〈 u, φ 〉 ∣ ≤ C ∑ ∣ α ∣ ≤ N sup ∣ ∂ α φ ∣. for all φ ∈ C c ∞ ( ℝ m ) with supports contained in K.
When the heat transfer is purely convective, or solely radiative, then one assumes that f is a linear functional (Newton's law of cooling), or obeys a fourth-order power law (Stefan's law), respectively.
Then, either F(A) = σUAU−1 + p(A) I, or F(A) = σUA′U−1 + p(A) I, where p is a linear functional on L X), U is a bounded linear bijective operator between the appropriate two spaces, σ is a complex constant, and A′ is the adjoint of A. The form of an operator F for which F and F−1 both send projections of rank one into projections of rank one is also determined.
Here α [ u ] is a linear functional on C [ 0, 1 ] given by α [ u ] = ∫ 0 1 u ( s ) d A ( s ).
