Can anybody explain about real linear space and complex linear space?

Question

This is definition

"A real linear $V$ is said to have an inner product if for each pair of elements $x$ and $y$ in $V$, there corresponds a unique real number $(x.y)$ satisfying what we know as axioms for inner product."

What does it mean? $(x.y)=xy$ and $xy$ is a real number?

When $(x.y)$ is complex, it is called complex space, does that mean that $(x.y)=xy=$ complex?

I don't understand what inner product is.

Answer 1

An inner product on a real vector space $V$ is a map $f : V\times V \to \mathbb{R}$ such that

• $f$ is symmetric (i.e. $f(x, y) = f(y, x)$),
• $f$ is linear in the first argument (i.e. $f(ax_1 + bx_2, y) = af(x_1, y) + bf(x_2, y)$ where $a, b \in \mathbb{R}$), and
• $f$ is positive definite (i.e. $f(x, x) \geq 0$ with equality if and only if $x = 0$).

Note, by symmetry, $f$ is also linear in the second variable, so $f$ is in fact bilinear.

Note that $f(x, y) \in \mathbb{R}$. Instead of calling the inner product $f$ and denoting its values by $f(x, y)$, it is common to instead label the function $(\cdot\,, \cdot)$ and denote its values by $(x, y)$. In particular, $(x, y)$ does not mean $xy$; the latter doesn't even make sense in general as you cannot multiply vectors.

An inner product on a complex vector space $V$ is a map $f : V\times V \to \mathbb{C}$ which is conjugate symmetric (i.e. $f(x, y) = \overline{f(y, x)}$), linear (over $\mathbb{C}$) in the first argument, and positive definite. In this case, $f$ is not linear in the second variable, but conjugate linear; that is, $f$ is sesquilinear.