# 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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### Norms Induced by Inner Products and the Parallelogram Law

Let $V$ be a normed vector space (over $\mathbb{R}$, say, for simplicity) with norm $\lVert\cdot\rVert$. It's not hard to show that if $\lVert \cdot \rVert = \sqrt{\langle \cdot, \cdot \rangle}$ ...
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### A norm which is symmetric enough is induced by an inner product?

$\newcommand{\<}{\langle} \newcommand{\>}{\rangle}$ It is a fact that for every norm $\| \|$ on a finite dimensional (real) vector space, its isometry group $\text{ISO}(|| \cdot ||)$ is ...
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### Show $(W_1 + W_2 )^\perp = W_1^\perp \cap W_2^\perp$ and $W_1^\perp + W_2^\perp ⊆ (W_1 \cap W_2 )^\perp$

Given $V$ a inner product space and $W_1$, $W_2$ subspaces of $V$ Show $(W_1 + W_2 )^\perp = W_1^\perp \cap W_2^\perp$ and $W_1^\perp + W_2^\perp ⊆ (W_1 \cap W_2 )^\perp$.
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### Why are every structures I study based on Real number?

I've been studying basic concepts of inner product vector space, normed vector space and metric space. And all the inner products, norms and metrics are defined to be real-valued functions in my ...
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### Do any books or articles develop basic Euclidean geometry from the perspective of “inner product affine spaces”?

Definitions. By a vector space, I simply mean an $\mathbb{R}$-module. By an affine space, I mean a vector space $X$ (the "translation space") together with a set $P$ (of "points"), together with an ...
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### In a complex vector space, $\langle Tx,x \rangle=0 \implies T = 0$

Suppose $T$ is a linear operator on a complex inner product space. Is it a theorem that if $\langle Tx,x\rangle=0$ for all $x$ in the space then $T=0$. The theorem fails in the real case, as seen for ...
### Rigid motion on $\mathbb{R}^2$ which fixes the origin is linear
Let $V=\mathbb{R}^2$ be an inner product space with the standard inner product, and let $T$ be a rigid motion of $V$. Suppose $T(0)=0$, prove that $T$ is linear. (A rigid motion of an inner product ...