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[家里蹲大学数学杂志]第036期泛函分析期末试题

2024-07-06 00:55:30   227人阅读

1 (15 分) 设 H<script id="MathJax-Element-1" type="math/tex">\mathcal{H}</script> 是 Hilbert 空间, l<script id="MathJax-Element-2" type="math/tex">l</script> 为 H<script id="MathJax-Element-3" type="math/tex">\mathcal{H}</script> 上的一实值线性有界泛函, C<script id="MathJax-Element-4" type="math/tex">C</script> 是 H<script id="MathJax-Element-5" type="math/tex">\mathcal{H}</script> 中一闭凸子集,

f(v)=12||v||2?l(v)(? v∈C).
<script id="MathJax-Element-6" type="math/tex; mode=display"> f(v)=\frac{1}{2}||v||^2-l(v)\quad(\forall\ v\in C). </script> 求证:

(1) 对任意 H<script id="MathJax-Element-7" type="math/tex">\mathcal{H}</script> 上线性有界泛函 g<script id="MathJax-Element-8" type="math/tex">g</script>, ? u0∈H<script id="MathJax-Element-9" type="math/tex">\exists\ u_0\in \mathcal{H}</script>, 使得 f(u0)=g(u0)<script id="MathJax-Element-10" type="math/tex">f(u_0)=g(u_0)</script>;

(2)? u1∈C<script id="MathJax-Element-11" type="math/tex">\exists\ u_1\in C</script>, 使得

f(u2)=infv∈Cf(v);
<script id="MathJax-Element-12" type="math/tex; mode=display"> f(u_2)=\inf_{v\in C}f(v); </script>

(3)讨论 g, u0, u1<script id="MathJax-Element-13" type="math/tex">g,\ u_0,\ u_1</script> 之间的关系.

 

2(15 分) 设 H<script id="MathJax-Element-14" type="math/tex">\mathcal{H}</script> 是 Hilbert 空间, T:H→H<script id="MathJax-Element-15" type="math/tex">T:\mathcal{H}\to \mathcal{H}</script> 是线性算子且满足

(Tx,y)=(x,Ty)(? x,y∈H).
<script id="MathJax-Element-16" type="math/tex; mode=display"> (Tx,y)=(x,Ty)\quad (\forall\ x,y\in \mathcal{H}). </script> 求证:

(1)T∈L(H)<script id="MathJax-Element-17" type="math/tex">T\in \mathcal{L}(\mathcal{H})</script>;

(2)T?=T<script id="MathJax-Element-18" type="math/tex">T^*=T</script>, 此时称 T<script id="MathJax-Element-19" type="math/tex">T</script> 为自共轭算子;

(3)若 R(A)ˉˉˉˉˉˉˉˉ=H<script id="MathJax-Element-20" type="math/tex">\overline{R(A)}=\mathcal{H}</script>, 则对 ? y∈R(A)<script id="MathJax-Element-21" type="math/tex">\forall\ y\in R(A)</script>, 方程

Ax=y
<script id="MathJax-Element-22" type="math/tex; mode=display"> Ax=y </script> 存在唯一解.

 

3(15 分) 证明:

(1)若 p≤q<script id="MathJax-Element-23" type="math/tex">p\leq q</script>, 则 lp?lq<script id="MathJax-Element-24" type="math/tex">l^p\subset l^q</script>;

(2)l∞<script id="MathJax-Element-25" type="math/tex">l^\infty</script> 不可分;

(3)l1<script id="MathJax-Element-26" type="math/tex">l^1</script> 不自反.

 

4(10 分) 设 φ∈C[0,1]<script id="MathJax-Element-27" type="math/tex">\varphi\in C[0,1]</script>, T: L2[0,1]→L2[0,1]<script id="MathJax-Element-28" type="math/tex">T:\ L^2[0,1]\to L^2[0,1]</script> 是由

(Tf)(x)=φ(x)∫10φ(t)f(t) dt(? f∈L2[0,1])
<script id="MathJax-Element-29" type="math/tex; mode=display"> (Tf)(x)=\varphi(x)\int_0^1\varphi(t)f(t)\ dt\quad(\forall\ f\in L^2[0,1]) </script> 给出的线性算子. 求证:

(1)T<script id="MathJax-Element-30" type="math/tex">T</script> 是自共轭算子 (定义见题2);

(2)? λ≥0<script id="MathJax-Element-31" type="math/tex">\exists\ \lambda\geq 0</script>, 使得 T2=λT<script id="MathJax-Element-32" type="math/tex">T^2=\lambda T</script>, 由此求出 T<script id="MathJax-Element-33" type="math/tex">T</script> 的谱半径 rσ(T)<script id="MathJax-Element-34" type="math/tex">r_\sigma(T)</script>.

 

 

5(10 分) 设 X<script id="MathJax-Element-35" type="math/tex">\mathcal{X}</script> 是自反的 Banach 空间, A?X<script id="MathJax-Element-36" type="math/tex">A\subset \mathcal{X}</script>. 证明:

(1)A<script id="MathJax-Element-37" type="math/tex">A</script> 弱列紧的充分必要条件是 A<script id="MathJax-Element-38" type="math/tex">A</script> 有界;

(2) 若 A<script id="MathJax-Element-39" type="math/tex">A</script> 弱列紧的, 则 A<script id="MathJax-Element-40" type="math/tex">A</script> 的凸包

co(A)={∑i=1nλixi; ∑i=1nλi=1, λi≥0, xi∈A, i=1,2,?,n, n∈N}
<script id="MathJax-Element-41" type="math/tex; mode=display"> co (A) =\left\{ \sum_{i=1}^n\lambda_ix_i;\ \sum_{i=1}^n \lambda_i=1,\ \lambda_i\geq 0,\ x_i\in A,\ i=1,2,\cdots, n,\ n\in \mathbb{N} \right\} </script> 也是弱列紧的.

 

6(10 分) 证明:

(1)在 Hilbert 空间 H<script id="MathJax-Element-42" type="math/tex">\mathcal{H}</script> 中, xn→x0<script id="MathJax-Element-43" type="math/tex">x_n\to x_0</script> 的充分必要条件是

||xn||→||x0||,xn?x0;
<script id="MathJax-Element-44" type="math/tex; mode=display"> ||x_n||\to ||x_0||,\quad x_n\rightharpoonup x_0; </script>

(2)在 L2[0,1]<script id="MathJax-Element-45" type="math/tex">L^2[0,1]</script> 中, fn→f<script id="MathJax-Element-46" type="math/tex">f_n\to f</script> 的充分必要条件是

fn?f,f2n??f2.
<script id="MathJax-Element-47" type="math/tex; mode=display"> f_n\rightharpoonup f,\quad f_n^2\stackrel{*}{\rightharpoonup} f^2. </script>

 

7(8 分) 设 H<script id="MathJax-Element-48" type="math/tex">\mathcal{H}</script> 是 Hilbert 空间, H0<script id="MathJax-Element-49" type="math/tex">\mathcal{H}_0</script> 是 H<script id="MathJax-Element-50" type="math/tex">\mathcal{H}</script> 的闭线性子空间, f0<script id="MathJax-Element-51" type="math/tex">f_0</script> 是 H0<script id="MathJax-Element-52" type="math/tex">\mathcal{H}_0</script> 上的线性有界泛函. 证明: ? H<script id="MathJax-Element-53" type="math/tex">\exists\ \mathcal{H}</script> 上的线性有界泛函 f<script id="MathJax-Element-54" type="math/tex">f</script>, 使得

f(x)=f0(x)(? x∈H0),
<script id="MathJax-Element-55" type="math/tex; mode=display"> f(x)=f_0(x)\quad(\forall\ x\in \mathcal{H}_0), </script>
||f||=||f0||.
<script id="MathJax-Element-56" type="math/tex; mode=display"> ||f||=||f_0||. </script>

 

 

8(8 分) 设 X, Y<script id="MathJax-Element-57" type="math/tex">\mathcal{X},\ \mathcal{Y}</script> 是 Banach 空间, T<script id="MathJax-Element-58" type="math/tex">T</script> 是 X<script id="MathJax-Element-59" type="math/tex">\mathcal{X}</script> 到 Y<script id="MathJax-Element-60" type="math/tex">\mathcal{Y}</script> 的线性算子, 又设对 ? g∈Y?<script id="MathJax-Element-61" type="math/tex">\forall\ g\in \mathcal{Y}^*</script>, g(Tx)<script id="MathJax-Element-62" type="math/tex">g(Tx)</script> 是 X<script id="MathJax-Element-63" type="math/tex">\mathcal{X}</script> 上的线性有界泛函, 求证: T<script id="MathJax-Element-64" type="math/tex">T</script> 是连续的.

 

9(9 分) 设 C[a,b]<script id="MathJax-Element-65" type="math/tex">C[a,b]</script> 是连续函数空间, 赋以最大值范数

||x||∞=supt∈[a,b]|x(t)|(? x∈C[a,b]).
<script id="MathJax-Element-66" type="math/tex; mode=display"> ||x||_\infty =\sup_{t\in [a,b]} |x(t)|\quad (\forall\ x\in C[a,b]). </script> 设 {xn}?C[a,b]<script id="MathJax-Element-67" type="math/tex">\{x_n\}\subset C[a,b]</script> x∈C[a,b]<script id="MathJax-Element-68" type="math/tex">x\in C[a,b]</script>. 求证: xn?x<script id="MathJax-Element-69" type="math/tex">x_n\rightharpoonup x</script> 的充分必要条件是
limn→∞xn(t)=x(t),? t∈[a,b]∩Q,
<script id="MathJax-Element-70" type="math/tex; mode=display"> \lim_{n\to\infty}x_n(t)=x(t),\quad \forall\ t\in [a,b]\cap \mathbb{Q}, </script> 且
supn≥1||xn||∞<∞.
<script id="MathJax-Element-71" type="math/tex; mode=display"> \sup_{n\geq 1}||x_n||_\infty<\infty. </script>

 

应老师要求, 出了一份泛函分析期末试卷, 主要针对张恭庆泛函分析第二章. 自己写完后也感觉太难了. 不过还是保留了做个纪念. 下次修改后再发终结版.


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