Е. Деза_ М.М. Деза. Энциклопедический словарь расстояний (2008) (1185330), страница 18

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䂇ÁËÌÓÏÓÈ Ì‡ V ̇Á˚‚‡ÂÚÒfl ÌÂÓÚˈ‡ÚÂθ̇flÙÛÌ͈Ëfl || ⋅ || : → , Û‰Ó‚ÎÂÚ‚Ófl˛˘‡fl ÚÂÏ Ê ‡ÍÒËÓχÏ, ˜ÚÓ Ë ÌÓχ, Á‡ ËÒÍβ˜ÂÌËÂÏ ÌÂ‡‚ÂÌÒÚ‚‡ ÚÂÛ„ÓθÌË͇, ÍÓÚÓÓ Á‡ÏÂÌflÂÚÒfl ·ÓΠÒ··˚Ï ÛÒÎÓ‚ËÂÏ: ÒÛ˘ÂÒÚ‚ÛÂÚ ÍÓÌÒÚ‡ÌÚ‡ ë > 0, ڇ͇fl ˜ÚÓ ‰Îfl ‚ÒÂı x, y ∈ V ‚˚ÔÓÎÌflÂÚÒfl ÌÂ‡‚ÂÌÒÚ‚Ó|| x + y || ≤ C)|| x || + || y ||)(ÒÏ. èÓ˜ÚË-ÏÂÚË͇, „Î. 1). èËÏÂÓÏ Í‚‡ÁËÌÓÏËÓ‚‡ÌÌÓ„Ó ÔÓÒÚ‡ÌÒÚ‚‡, ÍÓÚÓÓ Ì fl‚ÎflÂÚÒfl ÌÓÏËÓ‚‡ÌÌ˚Ï, ÏÓÊÂÚ ÒÎÛÊËÚ¸ ÎÂ·Â„Ó‚Ó ÔÓÒÚ‡ÌÒÚ‚Ó L p (Ω) Ò0 < p < 1, ‚ ÍÓÚÓÓÏ Í‚‡ÁËÌÓχ Á‡‰‡ÂÚÒfl ͇Í|| f ||= (∫Ω | f ( x ) |pdx )1 / p , f ∈ L p (Ω).Ň̇ıÓ‚Ó ÔÓÒÚ‡ÌÒÚ‚ÓŇ̇ıÓ‚Ó ÔÓÒÚ‡ÌÒÚ‚Ó (ËÎË Ç-ÔÓÒÚ‡ÌÒÚ‚Ó) ÂÒÚ¸ ÔÓÎÌÓ ÏÂÚ˘ÂÒÍÓÂÔÓÒÚ‡ÌÒÚ‚Ó (V, || x – y||) ̇ ‚ÂÍÚÓÌÓÏ ÔÓÒÚ‡ÌÒÚ‚Â V Ò ÏÂÚËÍÓÈ ÌÓÏ˚ || x – y||.ùÍ‚Ë‚‡ÎÂÌÚÌÓ, ÓÌÓ fl‚ÎflÂÚÒfl ÔÓÎÌ˚Ï ÌÓÏËÓ‚‡ÌÌ˚Ï ÔÓÒÚ‡ÌÒÚ‚ÓÏ (V, || ⋅ ||).Ç ˝ÚÓÏ ÒÎÛ˜‡Â ÌÓχ || ⋅ || ̇ V ̇Á˚‚‡ÂÚÒfl ·‡Ì‡ıÓ‚ÓÈ ÌÓÏÓÈ.

èËÏÂ‡ÏË·‡Ì‡ıÓ‚˚ı ÔÓÒÚ‡ÌÒÚ‚ fl‚Îfl˛ÚÒfl:1) l pn - ÔÓÒÚ‡ÌÒÚ‚‡, l p∞ - ÔÓÒÚ‡ÌÒÚ‚‡, 1 ≤ p ≤ ∞, n ∈ ;80ó‡ÒÚ¸ I. å‡ÚÂχÚË͇ ‡ÒÒÚÓflÌËÈ2) ÔÓÒÚ‡ÌÒÚ‚Ó ë ÒıÓ‰fl˘ËıÒfl ˜ËÒÎÓ‚˚ı ÔÓÒΉӂ‡ÚÂθÌÓÒÚÂÈ Ò ÌÓÏÓÈ || x || == supn | x n |;3) ÔÓÒÚ‡ÌÒÚ‚Ó ë0 ˜ËÒÎÓ‚˚ı ÔÓÒΉӂ‡ÚÂθÌÓÒÚÂÈ, ÍÓÚÓ˚ ÒıÓ‰flÚÒfl Í ÌÛβ ÔÓÌÓÏ | x || = maxn | xn ||;4) ÔÓÒÚ‡ÌÒÚ‚Ó C[pa, b ] ,1 ≤ p ≤ ∞ ÌÂÔÂ˚‚Ì˚ı ÙÛÌ͈ËÈ Ì‡ [a, b] Ò L p -ÌÓÏÓÈ|| f || p = (b∫a1| f (t ) | p dt ) p ;5) ÔÓÒÚ‡ÌÒÚ‚Ó ëä ÌÂÔÂ˚‚Ì˚ı ÙÛÌ͈ËÈ Ì‡ ÍÓÏÔ‡ÍÚ ä Ò ÌÓÏÓÈ || f || == maxt∈K | f(t)|;6) ÔÓÒÚ‡ÌÒÚ‚Ó (C [a,b])n ÙÛÌ͈ËÈ Ì‡ [a, b] Ò ÌÂÔÂ˚‚Ì˚ÏË ÔÓËÁ‚Ó‰Ì˚ÏË ‰ÓÔÓfl‰Í‡ n ‚Íβ˜ËÚÂθÌÓ Ò ÌÓÏÓÈ || f ||n =∑ k = 0 max a ≤ t ≤ b | f (k ) (t ) |;n7) ÔÓÒÚ‡ÌÒÚ‚Ó Cn[I m] ‚ÒÂı ÙÛÌ͈ËÈ, ÓÔ‰ÂÎÂÌÌ˚ı ‚ m-ÏÂÌÓÏ ÍÛ·Â Ë ÌÂÔÂ˚‚ÌÓ ‰ËÙÙÂÂ̈ËÛÂÏ˚ı ‰Ó ÔÓfl‰Í‡ n ‚Íβ˜ËÚÂθÌÓ Ò ÌÓÏÓÈ ‡‚ÌÓÏÂÌÓÈÓ„‡Ì˘ÂÌÌÓÒÚË ‚Ó ‚ÒÂı ÔÓËÁ‚Ó‰Ì˚ı ÔÓfl‰Í‡ Ì ·Óθ¯Â, ˜ÂÏ n;8) ÔÓÒÚ‡ÌÒÚ‚Ó M [a,b] Ó„‡Ì˘ÂÌÌ˚ı ËÁÏÂËÏ˚ı ÙÛÌ͈ËÈ Ì‡ [a, b] Ò ÌÓÏÓÈ|| f ||= ess sup | f (t ) | = infsup | f (t ) |;e, µ ( e ) = 0 t ∈[ a, b ] \ ea≤t ≤b9) ÔÓÒÚ‡ÌÒÚ‚Ó Ä (∆) ÙÛÌ͈ËÈ, ÍÓÚÓ˚ fl‚Îfl˛ÚÒfl ‡Ì‡ÎËÚ˘ÂÒÍËÏË ‚ ÓÚÍ˚ÚÓωËÌ˘ÌÓÏ ‰ËÒÍ ∆ = {z ∈ : | z | < 1} Ë ÌÂÔÂ˚‚Ì˚ÏË ‚ Á‡Í˚ÚÓÏ ‰ËÒÍ ∆ ÒÌÓÏÓÈ || f ||= maxz ∈∆ | f ( z ) |;10) η„ӂ˚ ÔÓÒÚ‡ÌÒÚ‚‡ Lp(Ω), 1 ≤ p ≤ ∞;11) ÔÓÒÚ‡ÌÒÚ‚‡ ëÓ·Ó΂‡ Wk,p(Ω), Ω ⊂ n, 1 ≤ p ≤ ∞ ÙÛÌ͈ËÈ f ̇ Ω, Ú‡ÍËı ˜ÚÓf Ë Â ÔÓËÁ‚Ó‰Ì˚ ‚ÔÎÓÚ¸ ‰Ó ÌÂÍÓÚÓÓ„Ó ÔÓfl‰Í‡ k ËÏÂ˛Ú ÍÓ̘ÌÛ˛ Lp-ÌÓÏÛ, cÌÓÏÓÈ || f ||k , p =∑i = 0 || f (i) ||0 ;k12) ÔÓÒÚ‡ÌÒÚ‚Ó ÅÓ‡ Äê ÔÓ˜ÚË ÔÂËӉ˘ÂÒÍËı ÙÛÌ͈ËÈ Ò ÌÓÏÓÈ|| f || = sup | f (t ) | .– ∞< t < +∞äÓ̘ÌÓÏÂÌÓ ‚¢ÂÒÚ‚ÂÌÌÓ ·‡Ì‡ıÓ‚Ó ÔÓÒÚ‡ÌÒÚ‚Ó Ì‡Á˚‚‡ÂÚÒfl ÔÓÒÚ‡ÌÒÚ‚ÓÏ åËÌÍÓ‚ÒÍÓ„Ó.

åÂÚË͇ ÌÓÏ˚ ÔÓÒÚ‡ÌÒÚ‚‡ åËÌÍÓ‚ÒÍÓ„Ó Ì‡Á˚‚‡ÂÚÒflÏÂÚËÍÓÈ åËÌÍÓ‚ÒÍÓ„Ó (ÒÏ. „Î. 6). Ç ˜‡ÒÚÌÓÒÚË, β·‡fl lp -ÏÂÚË͇ ÂÒÚ¸ ÏÂÚË͇åËÌÍÓ‚ÒÍÓ„Ó.ÇÒ n-ÏÂÌ˚ ·‡Ì‡ıÓ‚˚ ÔÓÒÚ‡ÌÒÚ‚‡ fl‚Îfl˛ÚÒfl ÔÓÔ‡ÌÓ ËÁÓÏÓÙÌ˚ÏË: ËıÏÌÓÊÂÒÚ‚Ó ÒÚ‡ÌÓ‚ËÚÒfl ÍÓÏÔ‡ÍÚÌ˚Ï, ÂÒÎË ‚‚Ó‰ËÚÒfl ‡ÒÒÚÓflÌË Ň̇ı‡–å‡ÁÛ‡dBM(V, W) = ln infT || T || ⋅ || T –1 ||, „‰Â ËÌÙËÏÛÏ ·ÂÂÚÒfl ÔÓ ‚ÒÂÏ ÓÔÂ‡ÚÓ‡Ï, ÍÓÚÓ˚Â‡ÎËÁÛ˛Ú ËÁÓÏÓÙËÁÏ T : V → W.lp -ÏÂÚË͇lp -ÏÂÚË͇ dl p , 1 ≤ p ≤ ∞ ÂÒÚ¸ ÏÂÚË͇ ÌÓÏ˚ ̇ n (ËÎË Ì‡ n), ÓÔ‰ÂÎÂÌ̇fl͇Í|| x – y ||p ,„‰Â lp -ÌÓχ || ⋅ ||p Á‡‰‡ÂÚÒfl ͇Ín|| x || p = (∑ | xi |i =11p p) .81É·‚‡ 5.

åÂÚËÍË Ì‡ ÌÓÏËÓ‚‡ÌÌ˚ı ÒÚÛÍÚÛ‡ıÑÎfl p = ∞ Ï˚ ÔÓÎÛ˜‡ÂÏ || x ||∞ = lim p →∞p∑i =1 | xi | p = max1≤ i ≤ n | xi | . åÂÚ˘ÂÒÍÓÂnÔÓÒÚ‡ÌÒÚ‚Ó ( n , dl p ) ÒÓÍ‡˘ÂÌÌÓ Ó·ÓÁ̇˜‡ÂÚÒfl Í‡Í l pn Ë Ì‡Á˚‚‡ÂÚÒfl l pn ÔÓÒÚ‡ÌÒÚ‚ÓÏ.lp -ÏÂÚË͇, 1 ≤ p ≤ ∞ ̇ ÏÌÓÊÂÒÚ‚Â ‚ÒÂı ÔÓÒΉӂ‡ÚÂθÌÓÒÚÂÈ x = {x n}∞n =1 ‰ÂÈÒÚ‚ËÚÂθÌ˚ı (ÍÓÏÔÎÂÍÒÌ˚ı) ˜ËÒÂÎ, ‰Îfl ÍÓÚÓ˚ı ÒÛÏχËÏÂÂÚ ‚ˉ∑ i =1 | x i | p∞(‰Îfl p = ∞ ÒÛÏχ∑i =1 | xi |) fl‚ÎflÂÚÒfl ÍÓ̘ÌÓÈ, ÓÔ‰ÂÎflÂÚÒfl ͇Í∞∞(∑ | xi − yi |1p p) .i =1ÑÎfl p = ∞ ÔÓÎÛ˜‡ÂÏ maxi≥1|xi – yi |.

ùÚÓ ÏÂÚ˘ÂÒÍÓ ÔÓÒÚ‡ÌÒÚ‚Ó ÒÓÍ‡˘ÂÌÌÓÓ·ÓÁ̇˜‡ÂÚÒfl Í‡Í l p∞ Ë Ì‡Á˚‚‡ÂÚÒfl l p∞ -ÔÓÒÚ‡ÌÒÚ‚ÓÏ.ç‡Ë·ÓΠ‚‡ÊÌ˚ÏË fl‚Îfl˛ÚÒfl l1 –, l2- Ë l∞-ÏÂÚËÍË; l2 -ÏÂÚË͇ ̇ n ̇Á˚‚‡ÂÚÒflÚ‡ÍÊ ‚ÍÎˉӂÓÈ ÏÂÚËÍÓÈ. l2 -ÏÂÚË͇ ̇ ÏÌÓÊÂÒÚ‚Â ÔÓÒΉӂ‡ÚÂθÌÓÒÚÂÈ {x n }n‰ÂÈÒÚ‚ËÚÂθÌ˚ı (ÍÓÏÔÎÂÍÒÌ˚ı) ˜ËÒÂÎ, ‰Îfl ÍÓÚÓ˚ı∑i =1 | xi |2 < ∞, ËÁ‚ÂÒÚ̇ Ú‡ÍÊÂ∞Í‡Í „Ëθ·ÂÚÓ‚‡ ÏÂÚË͇. ç‡ ‚Ò lp -ÏÂÚËÍË ÒÓ‚Ô‡‰‡˛Ú Ò Ì‡ÚÛ‡Î¸ÌÓÈ ÏÂÚËÍÓÈ| x – y |.Ö‚ÍÎˉӂ‡ ÏÂÚË͇ւÍÎˉӂ‡ ÏÂÚË͇ (ËÎË ÔËÙ‡„ÓÓ‚Ó ‡ÒÒÚÓflÌËÂ, ‡ÒÒÚÓflÌË "Í‡Í ÎÂÚ‡ÂÚ‚ÓÓ̇") dE – ÏÂÚË͇ ̇ n, ÓÔ‰ÂÎÂÌ̇fl ͇Í|| x = y ||2 = ( x1 − y1 )2 + … + ( x n − yn )2 .ùÚÓ Ó·˚˜Ì‡fl l2 -ÏÂÚË͇ ̇ n.

åÂÚ˘ÂÒÍÓ ÔÓÒÚ‡ÌÒÚ‚Ó (n, dE), ÒÓÍ‡˘ÂÌÌÓ̇Á˚‚‡ÂÚÒfl ‚ÍÎˉӂ˚Ï ÔÓÒÚ‡ÌÒÚ‚ÓÏ (ËÎË ‚¢ÂÒÚ‚ÂÌÌ˚Ï Â‚ÍÎˉӂ˚Ï ÔÓÒÚ‡ÌÒÚ‚ÓÏ). àÌÓ„‰‡ ‚˚‡ÊÂÌËÂÏ "‚ÍÎË‰Ó‚Ó ÔÓÒÚ‡ÌÒÚ‚Ó" Ó·ÓÁ̇˜‡ÂÚÒfl ÚÂıÏÂÌ˚È ÒÎÛ˜‡È n = 3, ‚ ÔÓÚË‚Ó‚ÂÒ Â‚ÍÎˉӂÓÈ ÔÎÓÒÍÓÒÚË ‰Îfl n = 2. Ö‚ÍÎˉӂ‡Ôflχfl (ËÎË ‰ÂÈÒÚ‚ËÚÂθ̇fl ‚ÍÎˉӂ‡ Ôflχfl) ÔÓÎÛ˜‡ÂÚÒfl ÔË n = 1, Ú.Â. fl‚ÎflÂÚÒfl ÏÂÚ˘ÂÒÍËÏ ÔÓÒÚ‡ÌÒÚ‚ÓÏ (, | x – y |) Ò Ì‡ÚÛ‡Î¸ÌÓÈ ÏÂÚËÍÓÈ (ÒÏ. „Î. 12).Ç ‰ÂÈÒÚ‚ËÚÂθÌÓÒÚË n fl‚ÎflÂÚÒfl ÔÓÒÚ‡ÌÒÚ‚ÓÏ ÒÓ Ò͇ÎflÌ˚Ï ÔÓËÁ‚‰ÂÌËÂÏ (ˉ‡Ê „Ëθ·ÂÚÓ‚˚Ï ÔÓÒÚ‡ÌÒÚ‚ÓÏ), Ú.Â. dE(x, y) = || x – y || = || x – y ||2 == 〈 x − y, x − y 〉 , „‰Â 〈x, y〉 ÂÒÚ¸ Ò͇ÎflÌÓ ÔÓËÁ‚‰ÂÌË ̇ n, ÍÓÚÓÓ Ô‰ÒÚ‡‚ÎÂÌÓ‚ ÒÓÓÚ‚ÂÚÒÚ‚Û˛˘ËÏ Ó·‡ÁÓÏ ‚˚·‡ÌÌÓÈ ÒËÒÚÂÏ (‰Â͇ÚÓ‚˚) ÍÓÓ‰ËÌ‡Ú ÙÓÏÛÎÓÈ〈 x, y 〉 =gij xi yi , „‰Âgij xi yi . Ç Òڇ̉‡ÚÌÓÈ ÒËÒÚÂÏ ÍÓÓ‰ËÌ‡Ú ËÏÂÂÏ 〈 x, y 〉 =n,∑ i, j∑ i, jgij = 〈ei, ej〉 Ë ÏÂÚ˘ÂÒÍËÈ ÚÂÌÁÓ ((gij)) fl‚ÎflÂÚÒfl ÔÓÎÓÊËÚÂθÌÓ ÓÔ‰ÂÎÂÌÌÓÈÒËÏÏÂÚ˘ÌÓÈ n × n χÚˈÂÈ.Ç Ó·˘ÂÏ ÒÎÛ˜‡Â ‚ÍÎË‰Ó‚Ó ÔÓÒÚ‡ÌÒÚ‚Ó ÓÔ‰ÂÎflÂÚÒfl Í‡Í ÔÓÒÚ‡ÌÒÚ‚Ó,Ò‚ÓÈÒÚ‚‡ ÍÓÚÓÓ„Ó ÓÔËÒ˚‚‡˛ÚÒfl ‡ÍÒËÓχÏË Â‚ÍÎˉӂÓÈ „ÂÓÏÂÚËË.ìÌËÚ‡̇fl ÏÂÚË͇ìÌËÚ‡̇fl ÏÂÚË͇ (ËÎË ÍÓÏÔÎÂÍÒ̇fl ‚ÍÎˉӂ‡ ÏÂÚË͇) ÂÒÚ¸ l2 -ÏÂÚË͇ ̇n, ÓÔ‰ÂÎÂÌ̇fl ͇Í|| x − y ||2 = | x1 − y1 |2 +…+ | x n − yn |2 .82ó‡ÒÚ¸ I.

å‡ÚÂχÚË͇ ‡ÒÒÚÓflÌËÈåÂÚ˘ÂÒÍÓ ÔÓÒÚ‡ÌÒÚ‚Ó (n, || x – y || 2 ) ̇Á˚‚‡ÂÚÒfl ÛÌËÚ‡Ì˚Ï ÔÓÒÚ‡ÌÒÚ‚ÓÏ (ËÎË ÍÓÏÔÎÂÍÒÌ˚Ï Â‚ÍÎˉӂ˚Ï ÔÓÒÚ‡ÌÒÚ‚ÓÏ). ÑÎfl n = 1 ÔÓÎÛ˜ËÏÍÓÏÔÎÂÍÒÌÛ˛ ÔÎÓÒÍÓÒÚ¸ (ËÎË ÔÎÓÒÍÓÒÚ¸ Ä„‡Ì‰‡), Ú.Â. ÏÂÚ˘ÂÒÍÓ ÔÓÒÚ‡ÌÒÚ‚Ó(, | z – u |) Ò ÏÂÚËÍÓÈ ÍÓÏÔÎÂÍÒÌÓ„Ó ÏÓ‰ÛÎfl | z – u |; | z | =| z1 + iz 2 |= z12 + z 22 Á‰ÂÒ¸fl‚ÎflÂÚÒfl ÍÓÏÔÎÂÍÒÌ˚Ï ÏÓ‰ÛÎÂÏ (ÒÏ. Ú‡ÍÊ ͂‡ÚÂÌËÓÌ̇fl ÏÂÚË͇, „Î. 12).Lp -ÏÂÚË͇Lp -ÏÂÚË͇ d L p , 1 ≤ p ≤ ∞ ÂÒÚ¸ ÏÂÚË͇ ÌÓÏ˚ ̇ L p (Ω, , µ), Á‡‰‡Ì̇fl ͇Í|| f – g ||p‰Îfl β·˚ı f, g ∈ L p (Ω, , µ). åÂÚ˘ÂÒÍÓ ÔÓÒÚ‡ÌÒÚ‚Ó ( L p (Ω, , µ ), d L p ) ̇Á˚‚‡ÂÚÒfl Lp -ÔÓÒÚ‡ÌÒÚ‚ÓÏ (ËÎË Î·„ӂ˚Ï ÔÓÒÚ‡ÌÒÚ‚ÓÏ).á‰ÂÒ¸ Ω – ÌÂÍÓÚÓÓ ÏÌÓÊÂÒÚ‚Ó Ë fl‚ÎflÂÚÒfl σ-‡Î„·ÓÈ ÔÓ‰ÏÌÓÊÂÒÚ‚ ÏÌÓÊÂÒÚ‚‡ Ω , Ú.Â.

ÒÂÏÂÈÒÚ‚ÓÏ ÔÓ‰ÏÌÓÊÂÒÚ‚ ÏÌÓÊÂÒÚ‚‡ Ω, Û‰Ó‚ÎÂÚ‚Ófl˛˘ËıÒÎÂ‰Û˛˘ËÏ Ò‚ÓÈÒÚ‚‡Ï:1) Ω ∈ ;2) ÂÒÎË A ∈ , ÚÓ Ω\A ∈ ;3) ÂÒÎË A = ∪ i∞=1 Ai c Ai ∈ , ÚÓ A ∈ .îÛÌ͈Ëfl µ : → ≥0 ̇Á˚‚‡ÂÚÒfl ÏÂÓÈ Ì‡ , ÂÒÎË Ó̇ ‡‰‰ËÚ˂̇, Ú.Â.µ(∪ i ≥1 Ai ) =µ( Ai ) ‰Îfl ‚ÒÂı ÔÓÔ‡ÌÓ ÌÂÔÂÂÒÂ͇˛˘ËıÒfl ÏÌÓÊÂÒÚ‚ A i ∈ ,∑ i ≥1Ë Û‰Ó‚ÎÂÚ‚ÓflÂÚ ÛÒÎӂ˲ µ(0/) = 0. èÓÒÚ‡ÌÒÚ‚Ó Ò ÏÂÓÈ Ó·ÓÁ̇˜‡ÂÚÒfl ÚÓÈÍÓÈ(Ω, , µ).ÑÎfl ‰‡ÌÌÓÈ ÙÛÌ͈ËË f : Ω → ()  Lp-ÌÓχ ÓÔ‰ÂÎflÂÚÒfl ͇Í|| f || p = 1∫Ωf (ω ) p µ( dω ) p .èÛÒÚ¸ L p (Ω, , µ) = L p (Ω) Ó·ÓÁ̇˜‡ÂÚ ÏÌÓÊÂÒÚ‚Ó ‚ÒÂı ÙÛÌ͈ËÈ f : Ω → (),ÍÓÚÓ˚ ۉӂÎÂÚ‚Ófl˛Ú ÛÒÎӂ˲ || f ||p < ∞. ëÚÓ„Ó „Ó‚Ófl, L p (Ω, , µ) ÒÓÒÚÓËÚ ËÁÍ·ÒÒÓ‚ ˝Í‚Ë‚‡ÎÂÌÚÌÓÒÚË ÙÛÌ͈ËÈ, „‰Â ‰‚ ÙÛÌ͈ËË ˝Í‚Ë‚‡ÎÂÌÚÌ˚, ÂÒÎË ÓÌËÔÓ˜ÚË ‚Ò˛‰Û Ó‰Ë̇ÍÓ‚˚, Ú. ÏÌÓÊÂÒÚ‚Ó, ̇ ÍÓÚÓÓÏ ÓÌË ‡Á΢‡˛ÚÒfl, ӷ·‰‡ÂÚÌÛ΂ÓÈ ÏÂÓÈ.

åÌÓÊÂÒÚ‚Ó L∞(Ω, , µ) ÂÒÚ¸ ÏÌÓÊÂÒÚ‚Ó Í·ÒÒÓ‚ ˝Í‚Ë‚‡ÎÂÌÚÌÓÒÚËËÁÏÂËÏ˚ı ÙÛÌ͈ËÈ f : Ω → (), ‡·ÒÓβÚÌ˚ ‚Â΢ËÌ˚ ÍÓÚÓ˚ı ÔÓ˜ÚË ‚Ò˛‰ÛÓ„‡Ì˘ÂÌ˚.ç‡Ë·ÓΠËÁ‚ÂÒÚÌ˚Ï ÔËÏÂÓÏ L p -ÏÂÚËÍË fl‚ÎflÂÚÒfl d L p ̇ ÏÌÓÊÂÒÚ‚ÂL p (Ω, , µ ), „‰Â Ω – ÓÚÍ˚Ú˚È ËÌÚÂ‚‡Î (0,1), – ·ÓÂ΂‡ σ-‡Î„·‡ ̇ (0,1) Ë µ –η„ӂ‡ ÏÂ‡. ùÚÓ ÏÂÚ˘ÂÒÍÓ ÔÓÒÚ‡ÌÒÚ‚Ó ÒÓÍ‡˘ÂÌÌÓ Ó·ÓÁ̇˜‡ÂÚÒfl Í‡Í Lp(0,1)Ë Ì‡Á˚‚‡ÂÚÒfl Lp(0,1)-ÔÓÒÚ‡ÌÒÚ‚ÓÏ.Ä̇Îӄ˘Ì˚Ï Ó·‡ÁÓÏ ÏÓÊÌÓ Á‡‰‡Ú¸ Lp-ÏÂÚËÍÛ Ì‡ ÏÌÓÊÂÒÚ‚Â C[ a, b ] ‚ÒÂı ‰ÂÈÒÚ‚ËÚÂθÌ˚ı (ÍÓÏÔÎÂÍÒÌ˚ı) ÌÂÔÂ˚‚Ì˚ı ÙÛÌ͈ËÈ Ì‡ [a, b]:bpd L p ( f , g) =  f ( x ) − g( x ) dx a∫1/ p.83É·‚‡ 5. åÂÚËÍË Ì‡ ÌÓÏËÓ‚‡ÌÌ˚ı ÒÚÛÍÚÛ‡ıÑÎfl p = ∞ d L∞ ( f , g) = max a ≤ x ≤ b | f ( x ) − g( x ) |. ùÚÓ ÏÂÚ˘ÂÒÍÓ ÔÓÒÚ‡ÌÒÚ‚Ó ÒÓÍ‡˘ÂÌÌÓ Ó·ÓÁ̇˜‡ÂÚÒfl Í‡Í C[pa, b ] Ë Ì‡Á˚‚‡ÂÚÒfl C[pa, b ] -ÔÓÒÚ‡ÌÒÚ‚ÓÏ.ÖÒÎË Ω = , = 2Ω fl‚ÎflÂÚÒfl ÒÂÏÂÈÒÚ‚ÓÏ ‚ÒÂı ÔÓ‰ÏÌÓÊÂÒÚ‚ Ω Ë µ – ͇‰Ë̇θÌ˚Ï ˜ËÒÎÓÏ (Ú.Â.

µ( A) = | A |, ÂÒÎË Ä – ÍÓ̘ÌÓ ÔÓ‰ÏÌÓÊÂÒÚ‚Ó Ω Ë µ(A) = ∞ –Ë̇˜Â), ÚÓ ÏÂÚ˘ÂÒÍÓ ÔÓÒÚ‡ÌÒÚ‚Ó( L (Ω, 2pΩ, | ⋅ | ), d L p)Ël∞p -ÔÓÒÚ‡ÌÒÚ‚ÓÒÓ‚-Ô‡‰‡˛Ú.ÖÒÎË Ω = Vn ÂÒÚ¸ ÏÌÓÊÂÒÚ‚Ó, ÒÓÒÚÓfl˘Â ËÁ n ˝ÎÂÏÂÌÚÓ‚, = 2 Vn , Ë µ fl‚ÎflÂÚÒfl(͇‰Ë̇θÌ˚Ï ˜ËÒÎÓÏ, ÚÓ ÏÂÚ˘ÂÒÍÓ ÔÓÒÚ‡ÌÒÚ‚Ó L p (Vn , 2 Vn , | ⋅ | ), d L p)Ël pn -ÔÓÒÚ‡ÌÒÚ‚Ó ÒÓ‚Ô‡‰‡˛Ú.Ñ‚ÓÈÒÚ‚ÂÌÌ˚ ÏÂÚËÍËlp -ÏÂÚË͇ Ë lq -ÏÂÚË͇, 1 < p , q < ∞ ̇Á˚‚‡˛ÚÒfl ‰‚ÓÈÒÚ‚ÂÌÌ˚ÏË, ÂÒÎË 1/p ++ 1/q = 1.Ç Ó·˘ÂÏ ÒÎÛ˜‡Â, ÍÓ„‰‡ ˜¸ ˉÂÚ Ó ÌÓÏËÓ‚‡ÌÌÓÏ ‚ÂÍÚÓÌÓÏ ÔÓÒÚ‡ÌÒÚ‚Â(V , || ⋅ ||V ), ËÌÚÂÂÒ Ô‰ÒÚ‡‚Îfl˛Ú ÌÂÔÂ˚‚Ì˚ ÎËÌÂÈÌ˚ ÙÛÌ͈ËÓ̇Î˚ ËÁ V ‚ÓÒÌÓ‚ÌÓ ÔÓΠ( ËÎË ).

ùÚË ÙÛÌ͈ËÓ̇Î˚ Ó·‡ÁÛ˛Ú ·‡Ì‡ıÓ‚Ó ÔÓÒÚ‡ÌÒÚ‚Ó(V ′, || ⋅ ||V ′ ), ̇Á˚‚‡ÂÏÓ ÌÂÔÂ˚‚Ì˚Ï ‰‚ÓÈÒÚ‚ÂÌÌ˚Ï ‰Îfl ÔÓÒÚ‡ÌÒÚ‚‡ V. çÓχ|| ⋅ ||V ′ ̇ V' Á‡‰‡ÂÚÒfl Í‡Í || T ||V ′ = sup|| x || ≤ 1 | T ( x ) |.( )çÂÔÂ˚‚Ì˚Ï ‰‚ÓÈÒÚ‚ÂÌÌ˚Ï ‰Îfl ÏÂÚ˘ÂÒÍÓ„Ó ÔÓÒÚ‡ÌÒÚ‚‡ l pn l p∞ fl‚ÎflÂÚÒfllqn(ÒÓÓÚ‚ÂÚÒÚ‚ÂÌÌÓl p∞ ).( )çÂÔÂ˚‚Ì˚Ï ‰‚ÓÈÒÚ‚ÂÌÌ˚Ï ‰Îfl ÔÓÒÚ‡ÌÒÚ‚‡ l1n l1∞l∞nl∞∞ ).fl‚ÎflÂÚÒfl(ÒÓÓÚ‚ÂÚÒÚ‚ÂÌÌÓçÂÔÂ˚‚Ì˚ ‰‚ÓÈÒÚ‚ÂÌÌ˚ ‰Îfl ·‡Ì‡ıÓ‚˚ı ÔÓÒÚ‡ÌÒÚ‚ ë (ÒÓÒÚÓfl˘Â„Ó ËÁ ‚ÒÂı ÒıÓ‰fl˘ËıÒfl ÔÓÒΉӂ‡ÚÂθÌÓÒÚÂÈ Ò lⴥ-ÏÂÚËÍÓÈ) ËC 0 (ÒÓÒÚÓfl˘Â„Ó ËÁ ‚ÒÂı ÒıÓ‰fl˘ËıÒfl Í ÌÛβ ÔÓÒΉӂ‡ÚÂθÌÓÒÚÂÈ Ò lⴥ-ÏÂÚËÍÓÈ)ÏÓ„ÛÚ ·˚Ú¸ ÂÒÚÂÒÚ‚ÂÌÌ˚Ï Ó·‡ÁÓÏ Ë‰ÂÌÚËÙˈËÓ‚‡Ì˚ Ò l1∞ .èÓÒÚ‡ÌÒÚ‚Ó ÒÓ Ò͇ÎflÌ˚Ï ÔÓËÁ‚‰ÂÌËÂÏèÓÒÚ‡ÌÒÚ‚ÓÏ ÒÓ Ò͇ÎflÌ˚Ï ÔÓËÁ‚‰ÂÌËÂÏ (ËÎË Ô‰„Ëθ·ÂÚÓ˚Ï ÔÓÒÚ‡ÌÒÚ‚ÓÏ) ̇Á˚‚‡ÂÚÒfl ÏÂÚ˘ÂÒÍÓ ÔÓÒÚ‡ÌÒÚ‚Ó (V , || x − y || ) ̇ ‰ÂÈÒÚ‚ËÚÂθÌÓÏ(ÍÓÏÔÎÂÍÒÌÓÏ) ‚ÂÍÚÓÌÓÏ ÔÓÒÚ‡ÌÒÚ‚Â V ÒÓ Ò͇ÎflÌ˚Ï ÔÓËÁ‚‰ÂÌËÂÏ 〈 x, y 〉Ú‡ÍÓ ˜ÚÓ ÏÂÚË͇ ÌÓÏ˚ || x − y || ÒÚÓËÚÒfl Ò ËÒÔÓθÁÓ‚‡ÌËÂÏ ÌÓÏ˚ Ò͇ÎflÌÓ„ÓÔÓËÁ‚‰ÂÌËfl || x || = 〈 x, x 〉 .ë͇ÎflÌÓ ÔÓËÁ‚‰ÂÌË 〈 , 〉 ̇ ‰ÂÈÒÚ‚ËÚÂθÌÓÏ (ÍÓÏÔÎÂÍÒÌÓÏ) ‚ÂÍÚÓÌÓÏ ÔÓÒÚ‡ÌÒÚ‚Â V fl‚ÎflÂÚÒfl ÒËÏÏÂÚ˘ÌÓÈ ·ËÎËÌÂÈÌÓÈ (‚ ÍÓÏÔÎÂÍÒÌÓÏ ÒÎÛ˜‡Â ÔÓÎÛÚÓ‡ÎËÌÂÈÌÓÈ) ÙÓÏÓÈ Ì‡ V, Ú.Â.

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