A normed space underlies an inner product space if and only if it satisfies the parallelogram law, or equivalently, if its unit ball is an ellipsoid.
32.
Note that each of the following objects is a special case of the types preceding it : normed spaces, Euclidean spaces, and the real / complex numbers.
33.
If and are normed spaces, they are "'isomorphic normed spaces "'if there exists a linear bijection such that and its inverse are continuous.
34.
If and are normed spaces, they are "'isomorphic normed spaces "'if there exists a linear bijection such that and its inverse are continuous.
35.
In particular, every continuous linear functional on a subspace of a normed space can be continuously extended to the whole space, without increasing the norm of the functional.
36.
It was simply referred to as property ( H ) in a list of properties for normed spaces that starts with ( A ) and ends with ( H ).
37.
A " linear transformation " between topological vector spaces, for example normed spaces, may be bounded, for example, when the domain is finite-dimensional.
38.
If the center is a distinguished point that is considered to be the origin of, as in a normed space, it is not mentioned in the definition and notation.
39.
This finite-dimensional version generalizes to functions & thinsp; and taking values in a normed space which could be for example a sequence space or an inner product space.
40.
Is a normed space but not an inner product space, because this norm does not satisfy the parallelogram equality required of a norm to have an inner product associated with it.
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