Inner product in a Hilbert space producing real numbers

Question

If we have some vectors, we know $$\langle a | b\rangle=\langle b|a \rangle ^*$$ Then if we consider $$\langle a | a\rangle=\langle a|a \rangle ^*$$

Then this tells us we will always get a real number. But why? In a Hilbert space, are the vectors all imaginary? Meaning that when you take the inner product (dot product) between two imaginary vectors they produce real numbers? Strictly speaking, does this real number need to be positive? My textbook says "The space $\mathcal{H}$ is endowed with a positive-definite scalar product, which makes it a Hilbert space", does positive-definite scalar product just mean the product is real or does it also mean it is positive?

Also why doesnt $$\langle a | a\rangle=\langle a|a \rangle ^*=0$$ If you take the vector $|a\rangle$ then its bra and ket elements are orthogonal, shouldn't the dot product between two orthogonal vectors give zero? My textbook says the above is only the case when the vector is zero.

Answer 1

Inner product space is a vector space V over the field F together with an inner product, that satisfies the following 3 properties:

1. $ \langle a | b\rangle=\langle b|a \rangle ^* $
2. $ \langle l \cdot a + m \cdot b | c\rangle=l \cdot \langle a|c \rangle + m \cdot \langle b|c \rangle $
3. If a is not 0; then: $ \langle a | a\rangle > 0 $

Now, I will answer your questions:

In a Hilbert space, are the vectors all imaginary?

There is no such thing as an imaginary vector. Did you mean a vector with imaginary components when written in some basis? Then the answer is obviously NO.

the inner product (dot product) between two imaginary vectors they produce real numbers?

Yes, the inner product between 2 vectors with purely imaginary components will be a real number.

Strictly speaking, does this real number need to be positive?

No, it can be any real number.

does positive-definite scalar product just mean the product is real or does it also mean it is positive?

It means that the inner product of a non-zero vector with itself, is a positive real number.

If you take the vector |a⟩ then its bra and ket elements are orthogonal

Nope. That is impossible.