Tags:definition |space

Inner Product

Definition

Definition. An inner product space is a vector space $V$ over the field $F$ together with an inner product, that is a function $⟨\cdot ,\cdot ⟩:V\times V\to F$ that satisfies the following three properties for all vectors $x,y,z\in V$ and all scalars $a,b\in F$

• Conjugate symmetry: $⟨x,y⟩=\overline{⟨y,x⟩}$. As $a=\overline{a}$ if and only if $a\in \mathbb{R}$, conjugate symmetry implies that $⟨x,x⟩$ is always a real number. If $F=\mathbb{R}$ conjugate symmetry is just symmetry.
• Linearity: $⟨ax+by,z⟩=a⟨x,z⟩+b⟨y,z⟩$.
• Positive-definiteness: if $x\ne 0$, then $⟨x,x⟩>0$.

The inner product induces a norm. Thus, an inner product space is also a normed space.

References

https://en.wikipedia.org/wiki/Inner_product_space