# Questions tagged [inner-product-space]

An inner product space is a vector space equipped with an inner product. The inner product is a generalization of the “dot” product often used in vector calculus.

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### Can the eigenvectors of a linear operator in an infinite-dimensional space span the space and be linearly dependent at the same time?

Consider a vector space $V$ over the complex field which is infinite-dimensional with a Euclidean inner-product. Let $L$ be a linear operator on $V$. Say a subset of eigenvectors of $L$ forms a ...
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### SVD with non-standard inner product

I have a linear transformation $T: V \to W$ where $V$ and $W$ are finitely generated real inner product spaces and their inner product is not necessarily standard. I also have $K: V \to V$. My goal is ...
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### Finding algebraic expression of a parallepiped given directions and side lengths

Let $u_1,\dots,u_n$ be linearly independent unit vectors in $\mathbb R^n$. Let $T$ be the parallepiped centred at $0$, with sides parallel to $u_i$ and side lengths $l_i$, $i=1,2,\dots,n$. My question ...
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### Prove the eigenvectors of a reflection transformation are orthogonal

This is part of a problem to prove that all reflection transformations are diagonalizable. For $T : \mathcal{V} \to \mathcal{V} \iff T^2 = I$, I've shown (1) that cases where $T$ has only one ...
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### Differential Equation by Minimization

Suppose we want to solve $u + xu' = 0$, which has the general solution $u = \frac{C}{x}$, by minimizing the length squared of $u + xu'$. This should work due to the positive definite condition of ...
### For $\langle Tx, x \rangle \geq \|x\|^2$, prove a solution exists for $Tx = y$.
Edit I've posted this a couple other times, so I now plan on deleting those, and just using this one. Here's the original problem: $H$ is a real Hilbert space. Let $T: H\longrightarrow H$ be a ...