An important theorem about continuous linear functionals on normed vector spaces is the Hahn Banach theorem.
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Its vectors form an inner product space ( in fact a Hilbert space ), and a normed vector space.
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When speaking of normed vector spaces, we augment the notion of dual space to take the norm into account.
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A vector space endowed with a norm, such as the Euclidean space, is called a normed vector space.
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:Usually, normed vector space are meant to be vector spaces over the field of real or complex numbers.
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Consequently, for normed vector space ( and hence Banach spaces ) the Bourbaki Alaoglu theorem is equivalent to the Banach Alaoglu theorem.
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In the case of a normed vector space, the polar of a neighbourhood is closed and norm-bounded in the dual space.
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An inner product space is a normed vector space, and the inner product of a vector with itself is real and positive-definite.
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As this property is very useful in functional analysis, generalizations of normed vector spaces with this property are studied under the name locally convex spaces.
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In mathematical analysis, the "'Minkowski inequality "'establishes that the L " p " spaces are normed vector spaces.
