What Are Shankar's Inner Product Axioms in Quantum Mechanics?

[Identity]

I was reading "Principles of Quantum Mechanics" - Shankar, and I'm having trouble understanding the inner product. Can someone help me or link me to a site that explains it?

The axioms of the inner product are

1. $$\langle V|W\rangle = \langle W|V\rangle^*$$

2. $$\langle V|V\rangle \geq 0\ \ \ \ \ 0 \ \ iff\ \ |V\rangle = |0\rangle$$

3. $$\langle V|(a|W\rangle +b|Z\rangle ) \equiv \langle V|aW+bZ\rangle = a\langle V|W \rangle +b\langle V|Z \rangle$$

Given that $$|V\rangle$$ and $$|W \rangle$$ can be expressed in terms of their basis vectors,

$$|V \rangle = \sum_i v_i |i \rangle$$

$$|W \rangle = \sum_j w_j|j \rangle$$

Shankar says "we follow the axioms obeyed by the inner product to obtain"

$$\langle V|W \rangle = \sum_i \sum_j v_i^*w_j\langle i|j \rangle$$

I don't understand how this comes about?

thanks

 

[Newtime]

I was reading "Principles of Quantum Mechanics" - Shankar, and I'm having trouble understanding the inner product. Can someone help me or link me to a site that explains it?

The axioms of the inner product are

1. $$\langle V|W\rangle = \langle W|V\rangle^*$$

2. $$\langle V|V\rangle \geq 0\ \ \ \ \ 0 \ \ iff\ \ |V\rangle = |0\rangle$$

3. $$\langle V|(a|W\rangle +b|Z\rangle ) \equiv \langle V|aW+bZ\rangle = a\langle V|W \rangle +b\langle V|Z \rangle$$

Given that $$|V\rangle$$ and $$|W \rangle$$ can be expressed in terms of their basis vectors,

$$|V \rangle = \sum_i v_i |i \rangle$$

$$|W \rangle = \sum_j w_j|j \rangle$$

Shankar says "we follow the axioms obeyed by the inner product to obtain"

$$\langle V|W \rangle = \sum_i \sum_j v_i^*w_j\langle i|j \rangle$$

I don't understand how this comes about?

thanks

What is your mathematical background? In most basic abstract algebra courses and even some linear algebra courses you will go into inner products and inner product spaces.

 

[Identity]

Oh I just finished high school and I'm trying to occupying myself in the holidays

(actually nevermind I was able to get it with some help from a friend)