The norm induced by an inner product is $\|x\|=\sqrt{\langle x,x\rangle}$. In standard coordinates it is Euclidean length, $\sqrt{\sum_j|x_j|^2}$. It is nonnegative, vanishes only at zero, obeys $\|cx\|=|c|\|x\|$, and satisfies the triangle inequality. Other norms exist, and not every norm comes from an inner product.
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- Definition of norm/length of a vectorFor $x\in\mathbb R^n$ or $\mathbb C^n$, the Euclidean norm is $\ x\ 2=(\sum j x j ^2)^{1/2}$. For $(3,4)$ it is $5$;...
- Definition of unit vectorA unit vector has norm one. Any nonzero vector $x$ can be normalized to $u=x/\ x\ $, which has the same direction....
- The inner product of a vector with itself is the square of its norm/length.By definition of the induced norm, $\ x\ ^2=\langle x,x\rangle$. In standard coordinates this is the sum of squared...
- Definition of distanceThe distance between vectors is $d(x,y)=\ x-y\ $. It is symmetric, nonnegative, zero exactly when $x=y$, and...
- The triangle inequalityFor any norm, $\ x+y\ \le\ x\ +\ y\ $. For an inner-product norm, expand $\ x+y\ ^2$ and bound the cross term using...
- The Cauchy–Schwarz inequalityThe Cauchy–Schwarz inequality is $ \langle x,y\rangle \le\ x\ \ y\ $. Equality holds iff the vectors are linearly...
- Definition of length/norm of a vectorThe norm induced by an inner product is $\ v\ =\sqrt{\langle v,v\rangle}$. It obeys positivity, absolute homogeneity...
- Definition of distance between vectorsThe inner-product distance is $d(u,v)=\ u-v\ =\sqrt{\langle u-v,u-v\rangle}$. It measures separation and satisfies...
- The Cauchy-Schwarz inequalityFor any inner-product space, $ \langle u,v\rangle ^2\le\langle u,u\rangle\langle v,v\rangle$. Equality holds iff...
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