Lecture Notes in Functional Analysis
Lecture Notes in Functional Analysis
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泛函分析讲义
DOI：
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发表时间：
2012
期刊：
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通讯作者：
A. Bressan
中科院分区：
文献类型：
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作者：
A. Bressan
1.1.Banach空间定义1.1导论。设X是K-向量空间。泛函p：x→[0，+∞)称为半范数，如果(A)p(λx)=|λ|p(X)，∀λ∈K，x∈X，(B)p(x+y)≤p(X)+p(Y)，∀x，y∈X。设p是半范数，使得p(X)=0⇒x=0。那么，p是范数(用∥⋅∥表示)。定义1.3。(X，∥⋅∥)对称为赋范线性空间。引理1.4。每个赋范空间(X，∥⋅∥)都是度量空间(X，d)，其度量由d(x，y)=∥x−y∥给出。
1.1. Introducton to Banach Spaces Definition 1.1. Let X be a K–vector space. A functional p ∶ X → [0,+∞) is called a seminorm, if (a) p(λx) = ∣λ∣p(x), ∀λ ∈ K, x ∈X, (b) p(x + y) ≤ p(x) + p(y), ∀x, y ∈X. Definition 1.2. Let p be a seminorm such that p(x) = 0 ⇒ x = 0. Then, p is a norm (denoted by ∥ ⋅ ∥). Definition 1.3. A pair (X, ∥ ⋅ ∥) is called a normed linear space. Lemma 1.4. Each normed space (X, ∥ ⋅ ∥) is a metric space (X,d) with a metric given by d(x, y) = ∥x − y∥.