Complete Inner Product Space
Hilbert Space versus Banach Space?
Im confused on some definitions. So apparently there are sets with a metric that are called metric spaces…if these are furthermore vector spaces armed with an inner product as their metric, and they are complete (cauchy sequences converge in the space) then they are hilbert spaces ? But a Banach space is a complete normed vector spaces…this sounds exactly like what a hilbert space is..help clarifying these concepts for me ?
The simplest distinction is that Hilbert spaces carry an inner product, but not Banach spaces. The inner product not only allows you to define the “length” (norm) of one individual vector, it also allows you to define the “angle” between two different vectors.
The distinction between a Banach space and Hilbert space is similar to the distinction between an arbitrary 4 dimensional vector space and R^4 with the Euclidean metric: in the former, there is not notion of orthogonality or inner product or projection that exists in the latter.
A normed space can be made into an inner product space IF AND ONLY IF the norm satisfies the parallelogram identity [1]. For an example of a normed vector space that is not an inner product space, consider the following:
Let V be the space of m-tuples of numbers (x1,x2,…,xm) with the norm that if X is an element in V, we write
|X| = |x1| + |x2| + … + |xm|
It is easy to verify that the norm |X| is nonnegative, that |X| = 0 implies X = (0,0,…,0), and that it satisfies the triangle inequality
|X+Y| <= |X| + |Y|
But it does not satisfy the parallelogram law. In fact, this is the norm for the so-called Manhattan Metric [2]. So V with the norm given above is not, and cannot be made into, an inner product space.
To summarise: a Hilbert space is a special type of Banach space. A Banach space can be made into a Hilbert space if the norm satisfies the parallelogram law.
Vectors – The Dot Product
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