# Questions tagged [inner-products]

For questions about inner products and inner product spaces, including questions about the dot product. An inner product space is a vector space equipped with an inner product. The dot product (seen in multivariable calculus and linear algebra) is a simple example of an inner product—other inner products may be seen as generalizations of the dot product.

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### Is there a equivalent norm on $L^p$ induced by inner product?

Suppose $L^p[a,b]$ is the normed space with the usual norm $\|f\|_p=(\int_a^b|f(x)|^p\mathrm{d}x)^{1/p}$. By the parallelogram equality, we know the norm is induced by an inner product iff $p=2$. ...
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### Properties of conformal maps

A linear map $T:X\to Y$ is said to be conformal when it preserves orthogonality; $\forall x,\tilde x\in X, \langle x,{\tilde x}\rangle =0\iff\langle{Tx},{T\tilde x}\rangle=0.$ Show that if $T$ is ...
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### A matrix with inner product that respects norm is a reflection matrix

Let $\langle \cdot | \cdot\rangle$ be the standard inner product over $\mathbb{R}^{2}$ and let $A \in M_{2}(\mathbb{R})$ be a matrix such that $\|Av\| = \|v\|$ for every $v \in \mathbb{R^{2}}$ and ...
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