Continuous and bounded linear functions and seminorms Every continuous linear operator is a bounded linear operator and if dealing only with normed spaces then the converse is also true. That is, a linear operator between two normed spaces is bounded if and only if it is a continuous function. So in particular, because the scalar field (which is or ) is a normed space, a linear functional on a normed space is a bounded linear functional if and only if it is a continuous linear functional. This allows for continuity-related results (like those below) to be applied to Banach spaces. Although boundedness is the same as continuity for linear maps between normed spaces, the term "bounded" is more commonly used when dealing primarily with Banach spaces.
Continuous and bounded linear functions and seminorms Every continuous linear operator is a bounded linear operator and if dealing only with normed spaces then the converse is also true. That is a linear operator between two normed spaces is bounded if and only if it is a continuous function. So in particular, because the scalar field (which is or ) is a normed space, a linear functional on a normed space is a bounded linear functional if and only if it is a continuous linear functional. This allows for continuity-related results (like those below) to be applied to Banach spaces. Although boundedness is the same as continuity for linear maps between normed spaces the term "bounded" is more commonly used when dealing primarily with Banach spaces