Elements of Functional Analysis and Distributions
The goal of this chapter is to recall, without proof, the main results in functional analysis: classical theorems about Fréchet, Hilbert, and Banach spaces, as well as fixed point theorems and an introduction to spectral theory. This is complemented by th
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Elements of Functional Analysis and Distributions
The goal of this chapter is to recall, without proof, the main results in functional analysis: classical theorems about Fréchet, Hilbert, and Banach spaces, as well as fixed point theorems and an introduction to spectral theory. This is complemented by the main definitions in distributions theory, including results about the Fourier transform.
1.1 Fréchet Spaces • Seminorms. Consider a vector space E on C. A seminorm on E is a map p : E → [0, +∞) such that, for all x, y ∈ E and all λ ∈ C, (i)
p(λx) = |λ|p(x),
(ii) p(x + y) ≤ p(x) + p(y).
A family P of seminorms on E is called separating if for every x ∈ E there exists p ∈ P such that p(x) = 0. • Topology. Let P be a family of separating seminorms on a vector space E. For x0 ∈ E, n ∈ N, n ≥ 1 and p ∈ P, set 1 . V(p,n) (x0 ) = x ∈ E : p(x − x0 ) < n Let Vx0 be the collection of all finite intersections of the sets V(p,n) (x0 ) and define a neighborhood of x0 as a set which contains an element of Vx0 . This defines a topology on E. Then (E, P) is called a locally convex space.
© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 T. Alazard, C. Zuily, Tools and Problems in Partial Differential Equations, Universitext, https://doi.org/10.1007/978-3-030-50284-3_1
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1 Elements of Functional Analysis and Distributions
– A subset A of E is said to be bounded if, ∀p ∈ P,
∃M > 0 : p(x) ≤ M,
∀x ∈ A.
– If (E, P) and (F, Q) are two locally convex spaces and T : E → F is a linear map then T is continuous if and only if, ∀q ∈ Q, ∃P0 ⊂ P finite, ∃C > 0 : q(T x) ≤ C
p(x), ∀x ∈ E.
p∈P0
– If P = (pj )j ∈N is countable then the topology defined above is metrizable, that is there exists a metric d on E which induces the same topology. Indeed for x, y ∈ E we set, d(x, y) =
+∞ 1 pj (x − y) · 2j 1 + pj (x − y) j =0
If moreover the metric space (E, d) is complete then we say that E is a Fréchet space. • Examples. – A Banach space is a Fréchet space. – Let be an open subset of Rd . Then C 0 () denotes the space of continuous functions on with complex values, C 1 () denotes the space of differentiable functions whose partial derivatives belong to C 0 (), and, for k ∈ N, k ≥ 2, ∂u C k () = u ∈ C k−1 () : ∈ C k−1 (), 1 ≤ j ≤ d . ∂xj Recall that can be written as = ∪+∞ j =0 Kj , where the Kj ’s are compact and Kj ⊂ Kj +1 . For k ∈ N ∪ {+∞} , u ∈ C k () and j ∈ N set, pj (u) =
sup |∂ α u(x)|
(k < +∞),
|α|≤k x∈Kj
pj (u) =
sup |∂ α u(x)|
(k = +∞).
|α|≤j x∈Kj
Then (C k (), (pj )j ∈N ) is a Fréchet space. A sequence (fn )n∈N ⊂ C k () converges to f for this topology if and only if, for every |α| ≤ k, (∂ α fn )n∈N converges to ∂ α f uniformly on each compact subset of .
1.2 Elements of Functional Analysis
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1.2 Elements of Functional Analysis In this section we recall several important results of functional analysis used in the study of partial differential equations. Unless expressly stated all the normed spaces considere
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