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maximal ideal space 是一個結合algebra 和 functional analysis 的concept. 設A為一Banach algebra，M為A的maximal ideal 的集合。可證M與A的multiplicative linear functionals有一一對應。由於A的multiplicative linear functionals 是A* 的子集，而A*上有Weak* topology，故M上有一induced topology，使M成為一topological space.這個space就稱作A的maximal ideal space.

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