# Questions tagged [inner-products]

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

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### The distance in $\mathbb{R}^n$ is not induced by an inner product.

We define the distance like $d(v,w)$ = {the number of different entries} and I'm supposed to prove that this is not induced by an inner product. So far what I´ve done is use the parallelogram law to ...
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### If $T$ is self-adjoint and $\alpha \in \mathbb{R}$, show that $\alpha \langle Tv,v \rangle \geq -|\alpha| \lVert Tv \rVert \lVert v \rVert$

I'm studying Sheldon Axler's "Liner Algebra done right" book, but I'm having some trouble understanding the proof of Lemma 7.11. Lemma (7.11): Suppose that $T \in \mathcal{L}(V)$ is self-...
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### Any inner product is determined by the lengths of vectors

For any complex inner product space $V$ and for any $u,v \in V$, we have: $$4 \langle u,v \rangle = \| u+v\| ^2 - \| u-v\|^2 +i\| u+iv\|^2 -i\| u-iv\|^2$$ to which my lecture notes conclude: "any ...
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### Comparison of the expectation of minimums of random variables

I'm examining the relationship between the expected values of the minimums of three independent, non-negative random variables $X$, $Y$, and $Z$, such that $E[2X] = E[Y] + E[Z]$. Specifically, I want ...
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### Find the matrix maximizing a summation of bilinear forms

Given $d, n \in \mathbb{N}$ and $x_k, y_k \in \mathbb{R}^d$, for $1 \leq k \leq n$, we want to find $\arg \max_{A: ||A||_2 = 1} \sum_k \langle x_k, A y_k \rangle$, where $||\cdot||_2$ denotes spectral ...
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### $T$ and $U$ self-adjoint on $V$, $T$ is positive definite. Prove $TU$ & $UT$ are diagonalizable linear operators that have only real eigenvalues.
This is a problem from Freidberg linear algebra (4th edition) chapter 6.4.21 The whole problem and the hints are Let $V$ be a finite-dimensional inner product space, and let $T$ and $U$ be self-...