I'm stuck with the following exercise. Hopefully some of you can help me.
Exercise
A normed K-vectorspace is a $\mathbb{K}$ -vectorspace $V,$ where a transformation is defined as:
$\|\cdot\| : V \rightarrow[0, \infty[$
for all $v, w \in V$, where the following conditions has to be met:
a) $\|v\|=0$ if and $\mathrm{only}$ if $v=0 .$
b) $\|\alpha \cdot v\|=|\alpha| \cdot\|v\|$ for $\alpha \in \mathbb{K}$
c) $\|v+w\| \leq\|v\|+\|w\|$
1) Show, that the norm on a inner product space $V$ makes $V$ to a normed vectorspace.
In the following questions is considered the real vectorspace $V=\mathbb{R}^{2}$ and the following transformation:
$$\|\cdot\| : \mathbb{R}^{2} \rightarrow[0, \infty[$$
$$\quad \left( \begin{array}{l}{\alpha} \\ {\beta}\end{array}\right) \mapsto|\alpha|+|\beta|$$
We define:
$\langle\boldsymbol{v}, \boldsymbol{w}\rangle=\frac{1}{4}\left(\|\boldsymbol{v}+\boldsymbol{w}\|^{2}-\|\boldsymbol{v}-\boldsymbol{w}\|^{2}\right)$
for all $\boldsymbol{v}, \boldsymbol{w} \in \mathbb{R}^{2}$
5) Conclude, that the transformation $\|\cdot\|$ above, is not defined from the inner product on $V=\mathbb{R}^{2}($ Hint: Polarization identity $)$
My approach
1) What am I supposed to do here?
5) Here is my approach:
$|\alpha|+|\beta|=\|v\|=(<v, v>)^{\frac{1}{2}}=\left(\frac{1}{4}\left(\|v+v\|^{2}-\|v-v\|^{2}\right)\right)^{\frac{1}{2}}=\left(\frac{1}{4}\|2 v\|^{2}\right)^{\frac{1}{2}}=\left(\|v\|^{2}\right)^{\frac{1}{2}}=\|v\|$
So we can see that the transformation is defined by $\langle\boldsymbol{v}, \boldsymbol{w}\rangle$, but we just showed above in 4) that $\langle\boldsymbol{v}, \boldsymbol{w}\rangle$ is not an inner product.
I will answer your first question:
Every inner product $\langle \cdot,\cdot\rangle$ induces a norm $\left\lVert\cdot\right\rVert$ via $$\left\lVert x\right\rVert=\sqrt{\langle x,x\rangle}.$$ They want you to prove that this is, indeed a norm i.e. that the above function satisfies the norm properties.
I'm not sure what your question is for 5).