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過去の入試問題 | Department of Mathematics Kyoto University
6
(1)はルベーグの収束定理を使うためにうまい優関数を見つける必要があります. やること自体はシンプルなのですが何となく馴染みのない計算で少し苦労しました. (2)は特段詰まるところのない問題です.
(1)
Define .
From an elementary calculation, we have for
. Therefore it follows that for
,
(1)
Since the function




(2)
Define . Fix
and take
so that
. Since we have
,
it holds that
(2)
Hence for every



![Rendered by QuickLaTeX.com x \in [x_0 - \delta, x_0 + \delta]](http://paleperlite.com/wp-content/ql-cache/quicklatex.com-b27c7b39bd4e4fba1a9f0b791159f9c1_l3.png)



7
(1)はアスコリ-アルツェラの定理を使ってコンパクト性を示します. 定番の方法といえるでしょう.
(2)は具体的に共役作用素が求まることがポイントです. 最終的には常微分方程式の境界値問題になります.
(1)
Let be a sequence in the unit ball of
.
From Arzela-Ascoli theorem, it is enough to show that the sequence is uniformly bounded and equicontinuous in
.
From Hölder’s inequality, we have for every ,
(3)
where

![Rendered by QuickLaTeX.com L^2([0,1])](http://paleperlite.com/wp-content/ql-cache/quicklatex.com-120329f0cbd6d46f7911f558d5da7836_l3.png)
From this, the equicontinuity and uniform boundedness is obvious.
(2)
From an elementary calculation, we can check that
(4)
Since the operator

Take



(5)
Since the LHS of (5) is continuous, the function



Differentiating both sides twice in (5), we obtain
(6)
Since


(7)
If , from (6), it holds that
(8)
for constants




(9)
and we can easily check that this function satisfies (5).
If , it follows from (6) that
(10)
for constants



Therefore the eigenvalues of are
.