A Hilbert space is a vector space with an inner product such that the norm defined by
turns into a complete metric space. If the metric defined by the norm is not complete, then is instead known as an inner product space.
Examples of finite-dimensional Hilbert spaces include
1. The real numbers with the vector dot product of and .
2. The complex numbers with the vector dot product of and the complex conjugate of .
An example of an infinite-dimensional Hilbert space is , the set of all functions such that the integral of over the whole real line is finite. In this case, the inner product is
A Hilbert space is always a Banach space, but the converse need not hold.
A (small) joke told in the hallways of MIT ran, "Do you know Hilbert? No? Then what are you doing in his space?" (S. A. Vaughn, pers. comm., Jul. 31, 2005).