以下为句子列表：
英文: Abstract: In this paper,some inequalities which generalize the results of our predecessors are established in real or complex inner product spaces.
中文: 文摘:在实或复的内积空间中建立了几个不等式，推广了前人的工作.
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英文: Simulation results show that the number of inner product between the image or its residual image and atoms is only 4000 times in each calculaltion step, and the reconstructed image has fine visual quality.
中文: 仿真结果表明，每步分解所需计算的图像或图像残差与原子的内积仅4000次，由分解结果重建的图像具有较好的主观质量。
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英文: The inner product of the mapping value of the original data in feature space is replaced by a kernel function, and the weights of each neuron can be initialized and updated by initializing and updating the combinatorial coefficient vector of each weight i
中文: 该算法以核函数代替原始数据在特徵空间中映射值的内积，并且神经元权值向量的初始化和更新都可由其组合系数向量表示，从而获得了直观而简单的迭代公式。
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英文: There are many different types of abstract space in mathematics, for examples, vector space, inner product space, etc.
中文: 数学上存在各种不同的抽象空间，例如向量空间，内积空间等。
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