Inner Product

Verify that $\mathbb{R}^{n}$ along with dot product is an inner product space.

Show that Hermitian inner product on the two-dimensional vector space $\mathbb{C}^{2}$ satisfies the three conditions in the definition of inner product.

Show that this operation is an inner product on the real vector space $\mathcal{M}_{2\!\times\!2}$.

Consider the real vector space of quadratic polynomials $\mathcal{P}_{2}$. Show that this is an inner product operation: $\langle a_{2}x^{2}+a_{1}x+a_{0},b_{2}x^{2}+b_{1}x+b_{0}\rangle=a_{2}b_{2}+a_{1}b_{1}+a_{0}b_{0}$. Consider the resulting norm. What is $\lVert x^{2}-1\rVert$?

Verify that the set of real valued functions that are continuous on the closed interval $[\,0\ldots 1\,]$ is a vector space.

Show that an attempted definition of $\langle\vec{v},\vec{w}\rangle$ on $\mathbb{C}^{n}$ that is linear in both inputs loses that $\langle\vec{v},\vec{v}\rangle$ gives the length.