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1、Def 4.1 orthonormal subset H:1)e=1,e ;2)e e,e,e,e e.标准正交集是线性无关集：e+e+e=0 e+e+e,e=0 =0 Th4.8(Bessels inequality)设 H 是内积空间，e，e，是标准正交集合，h H.则|h,e|h.Pf.h h,ee h,ee,e =0 h h,ee h,ee,h,ee =0 0 h h,ee h,ee,h h,ee =h h,ee h,ee,h =h|h,e|h,e|h.|h,e|h.Th4.9 设 H 是内积空间，是标准正交集合，h H.则h,e 0 for at most a countable n
2、umber of vector e in.Pf.e :h,e 0=e :|h,e|e :|h,e|1n中至多有限个（Bessels inequality）e :h,e 0中至多可列个。内积空间肯定存在最大标准正交集。内积空间中的标准正交集都可以扩成最大标准正交集。（Th.4.2）Th4.14 Any two bases have the same cardinality.Th4.13 设 H 是 Hilbert 空间，是标准正交集合。则(1)0 h=h,ee,h H;(h=h,ee,h H)(2)0 h=|h,e|,h H.(Parsevals Indentity)(h=|h,e|,h H)证
3、明：(1)|h,e|h h,ee 的部分和是 Cauchy 列。（S S=h,ee,h,ee =|h,e|）H 是 Hilbert 空间，h,ee 存在。h h,ee,e(or e)=0 h h,ee .0 h h,ee=0.(2)h=h,h=h,ee,h,ee =limh,ee,h,ee =lim|h,e|=|h,e|.Def.A normed space X is called separable if it has a countable dense subset.Th4.16 设 H 是无穷维 Hilbert 空间，则 H is separable iff dimH=.Th4.6 Th
4、e Gram-Schmidt orthogonalization Process:If h,h,is a linearly independent subset of H,then there is an orthonormal set e,e,s.t.for every n,span h,h,=spane,e,.注：X a linear space,B a subset of X spanB b+b+b:,b B,n 1 注：设 X 是线性空间，X=span x,x,x,y,y,y是 X 中的线性无关集，则 m n.内积空间的标准正交基（orthonormal basis）和代数基（Hame
5、l basis）:1，无穷维 Hilbert 空间，最大标准正交集肯定不是最大线性无关集。2，无穷维 Hilbert 空间，最大线性无关集中元素个数.Def.5.1 If H,K are Hilbert spaces,there is a linear surjection U:H K s.t.Uh,Uh=h,hh,h H,then H and K are called to be isomorphic.Th 5.2 设 H,K 是内积空间，T:H K 是线性算子。则Th=h,h H iff Th,Th=h,hh,h H.Th 5.4 Two Hilbert spaces are isomor
6、phic iff they have the same dimension.For an arbitrary topological space,there is a generalization of the notion of sequence,called a“net”.Recall that a relation on a set A is called a partial order relation if the following conditions hold:(1)x x,x A(2)If x y and y x,then x=y(3)If x y and y z,then
7、x z.A directed set A is a set with a partial order such that for each pair x,y of A,there exists an element z of A having the property that x z and y z.(1)Show that if X is Hausdorff,a net in X converges to at most one point.(2)Let A X.Then x A if and only if there is a net of points of A converging
8、 to x.Hint:To prove the implication,take as index set the collection of all neighborhoods of x,partially ordered by reverse inclusion.(3)Let f:X Y.Then f is continuous if and only if for every convergent net(x)X,x x,the net f(x)f(x).Def.A subset K of A is said to be cofinal in A if for each x A,ther
9、e exists k K s.t.x k。Show that if A is a directed set and K is cofinal in A,then K is a directed set.Def.Let X be a topological space,J,K be directed set,g:K J,f:J X s.t.1)i j g(i)g(j);2)g(K)is cofinal in J then f g:K X is called a subnet of x(or f().Show that if the net(x)converges to x,so does any
10、 subnet.Def.Let X be a topological space,(x)be a net in X.We say that x is an accumulation point of the net(x)if for each neighborhood U of x,the set of those for which x U is cofinal in J.Show that the net(x)has the point x as an accumulation point if and only if some subnet of(x)converges to x.Def
11、.We say a topological space(X,)is compact if any open cover of X has a finite subcover,that is,for any family with X=U,there is a finite subset U,U,U with X=U.A subset of a topological space is called compact set if it is a compact space in the relative topology.Def.A topological space X is said to
12、have the finite intersection property(f.i.p.)if and only if any family of closed sets with F for any finite subfamily F,i=1,2,n satisfies F .Th.(f.i.p.criterion)X is compact if and only if X has the f.i.p.。Th.(The Bolzano-Weierstrass theorem)A space X is compact if and only if every net in X has a convergent subnet.注：First countable spaces are compact if and only if every sequence has a convergent subsequence.