内积空间

The content of this course is divided into four chapter, the normed linear space and bounded linear operator on the normed space; the character of finite dimensional linear space; the basic theorems on Banach space.

本课程主要分为四章，赋范线性空间与内积空间；赋范线性空间上的有界线性算子，有限维赋范线性空间的特征。Banach空间中的基本定理：泛函存在定理，一致有原理，开映象，闭图象、逆算子定理。

Hilbert space and generalized Fourier series in Hilbert space,etc.

Hilbert空间与广义Fourier级数，主要讲授内积空间内积空间中的正交系，投景定理，与广义Fourier级数。

Thesolutions compose self-contained inner product space (weight function is constant 1),i.e., they are basic series of Hilbert space; hence any internal waves could beexpanded into generalized Fourier series. With difference method, Sturm-Liouvilleequation is transformed to matrix eigenvalue problem. Each eigenvalue is related to ahorizontal wave number, and relevant solution is a certain mode of internal wave.

其解系构成完备的加权(权函数为1)内积空间，即Hilbert空间，从而内波现象可以展成该解系的广义Fourier级数；采用差分方法，将 Sturm-Liouville 方程转化为矩阵特征值问题，每一个特征值对应一个内波波数，该特征值对应的解则是内波的某一模态。

In the third chapter, it is proved that a Minkowski plane withπ/2-property, whose unit circle is Radon curve, is an inner product space.

在第三章中，本文证明了单位圆是Radon曲线且具有π/2性质的空间是内积空间。

Based on the minimizing vector theorem of the fuzzy inner product space ,the pseudo fuzzy orthogonal vector is defined and the projected theorem of the fuzzy inner product space is testified.

在模糊内积空间的极小化向量定理基础上，给出了向量拟模糊正交定义，并在模糊内积空间中证明了投影定理。

Firstly,the generic conception of symmetry for a real sequence was proposed,and then,the symmetry-decomposition and the symmetric degree sequence are presented,which are deduced from the projection theory,the orthogonal-decomposition theory in inner product space and .

首先提出序列信号一般意义下的对称概念，然后由内积空间中的投影、正交分解理论以及内积量化两个信号线性相关程度的特性导出任意信号的对称分解及对称程度序列，对称程度序列定量刻画了信号随对称点的变化时对称特性的变化，在此基础上得出任意序列信号对称程度的定量指标——对称性指标。

We paid special attention to the complex linear space consisting of all linear mappings from dimensional complex inner product space into dimensional complex inner product space .

2是与基的选择无关的概念。容易验证内积(2)满足内积的公理，从而是一个位的复内积空间。当是的正交归一基，是的正交归一基，可以证明构成的正交归一基。

We first analyze the relationship between the characteris-tic subspaces of linear transformation A,and its conjuge transformation A*, on then-dimensional complex inner product space Cn, and prove that the characteristicsubspace of an enginvalueλof A is orthogonal to that of enginvalue μ of A* if λ≠μ.

首先，分析n-维复内积空间Cn上的线性变换A与其共轭变换A*的特征子空间之间的关系，证明出只要A的一个特征值λ与A*的一个特征值μ的共轭不同，那么它们对应的特征子空间是正交的。

We found , however, that the concept of hyper-operator itself is very important for quantum physics , group representation and Lie algebras.

本文内容的具体安排如下：第一章介绍了线性空间，赋范线性空间，Banach 空间，内积空间和Hilbert 空间的概念。

In order to talk about the limitation of a vector series, one has to introduce the norm of a vector.

量子体系的态矢量空间是复的内积空间，即定义了内积的复线性空间。

But we don't care about Battlegrounds.

但我们并不在乎沙场中的显露。

Ah! don't mention it, the butcher's shop is a horror.

啊！不用提了。提到肉，真是糟透了。