# Questions tagged [inner-products]

For questions about inner products and inner product spaces, including questions about the dot product.

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### In a finite dimensional inner product space with $T ∈ L(V)$, show that $\langle u,v\rangle = \langle T(u),T(v)\rangle$ implies $T$ is invertible.

Here is how I've tried to go about it, and I'm curious if it's true or if I'm way off base. T is invertible iff null$(T)=\{0\}$. Let $v∈V$ and suppose $T(v)=0$. If we can show that $v=0$, then $T$ is ...
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### Divergence of Petersson inner product

Consider a complex number $z = x + i y$ and functions $f,g : \mathbb{H} \to \mathbb{C}$ (where $\mathbb{H}$ is the upper half-plane i.e. complex numbers whose imaginary part is greater equal to zero). ...
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### Application of Cauchy-Schwarz with Sobolev norms

I'm working through the problems in the initial value formulation chapter in Wald's General Relativity. A short summary of the problem. I have to show that $$\sup_{x\in A}|f(x)|\le C||f||_{A,k}$$ ...
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### Inner product alternative definition

I'm trying to make an alternative (but equivalent) definition of an inner product. I prefer to use arbitrary sums with $\sum_i v_i$ instead of sum of just two vectors $v+w$, and few but clear axioms. ...
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