Let be a real linear space. The mapping is called a "scalar product" if it has the following properties:

1. ,

2. The mapping is linear for each ,

3.

4. .

The real linear space equipped with a scalar product is called "Hilbert space" if it is a Banach space with respect to the norm .

(Riesz representation theorem). For each there exists a such that

(Lax-Milgram theorem). Let be a Hilbert space and be a bilinear mapping: for some constants . Fix an . There exists a unique such that