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Dołączył: 17 Gru 2010
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| Temat postu: Finite dimensional linear space Quasinorm derivati |
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Finite dimensional linear space Quasinorm derivative of vector function
The Ilfll 0, so that the three 'IBx) I> (I knock | Qian Mountain (f2II thirty {) = M, (I) Rights II thirty If'z-●. ln the l dagger available Mi, 22 finite-dimensional linear space Quasinorm derivative of vector function E 是 set, I D r th Quasinorm Linear Space. {e, e, ...,) is E, a base, take any x ∈ E , x = Σ Island, {f ∈ R. i = I in [Ⅱ, hi defined on the vector function with values in E: x (t) = Σ {(£)£ ∈ [d, hif = 1 where each component (f) is [6] on the real-valued function. Theorem 2 Let H E is the power of r-dimensional linear space Quasinorm, (£) is defined in [b] on the vector function with values in E. If x (£) in [d, hi weakly on the guide pin, then x (f) for each component. (r) in [d, b] j can be guided, and x (t) the weak derivative of x '(£)= Σ {。'(£)£ ∈ hi 【= _ Certificate: From the definition of weak derivative, for any £. ∈ hi, there is x '(c port) = Σ. ∈ E, so that for any a, ∈ E ·, there are so lim. A x ,[link widoczny dla zalogowanych],(£。)) 一 0 一 0, H. J1Jif, 10 ≠ ff, J = 1,2, ..., then 6f) ∈ R. corresponds to take. (F) can be guided in the r and. = '(£ o). Therefore, x '(a dictionary) = Σe. '(Go) e. . Lol = I by the arbitrary nature of race has】 ● 8 abnormal Zhu Rong Fungal Engine Fan palm University reported in 1992, x '(f) = ΣI' (f) iI Theorem 3 Let r E is the H-dimensional power of Quasinorm linear space, which caused a continuous sheet range, the E of the vector function x 【£) (f ∈ [mouth, b】) derivative of the weak and strong derivatives turn equivalent. Proof: easy to see that x (f) is the strong derivative of its weak derivative. The following license Blockhouse x (I) is the weak derivative of its strong derivative. Let x (r) = Σe. (T) in, on the weak to be found, JlI by Theorem 2. x '(t) = Σe. '(F) t ∈ £ lI'lb]. Because the E effect on the proposed scope of continuous, so by [4] Theorem 1, there exists M> 0, so that fI is called (t) ll <(f industry, a good o (n 一 0} (2J so x (f) in, b], Strong guidance, and strong derivative is equal to the weak derivative 'because of than in the finite dimensional Quasinorm linear space the strong derivative and weak derivative indiscriminately referred to as derivatives. Theorem 4 Let E be a finite dimensional r th Quasinorm Linear Space, which proposed a continuous range of drums, Ba V the vector function xff) = Σ. (t) t ∈ [n, 6】 to guide the necessary and sufficient condition is x (tl e of each component . (tl can jI guide. prove that: J need to be obtained from the Theorem 2, g points of the face of God by the theorem 3 (2) can be obtained .0 parlous: Bi Zhan Wang Dun-guided fork painting l9
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