Editing Normed vector space (section)

==Finite product spaces==

Given <math>n</math> seminormed spaces <math>\left(X_i, q_i\right)</math> with seminorms <math>q_i : X_i \to \R,</math> denote the [[product space]] by
<math display=block>X := \prod_{i=1}^n X_i</math>
where vector addition defined as
<math display=block>\left(x_1,\ldots,x_n\right) + \left(y_1,\ldots,y_n\right) := \left(x_1 + y_1, \ldots, x_n + y_n\right)</math>
and scalar multiplication defined as
<math display=block>\alpha \left(x_1,\ldots,x_n\right) := \left(\alpha x_1, \ldots, \alpha x_n\right).</math>

Define a new function <math>q : X \to \R</math> by
<math display=block>q\left(x_1,\ldots,x_n\right) := \sum_{i=1}^n q_i\left(x_i\right),</math>
which is a seminorm on <math>X.</math> The function <math>q</math> is a norm if and only if all <math>q_i</math> are norms.

More generally, for each real <math>p \geq 1</math> the map <math>q : X \to \R</math> defined by 
<math display=block>q\left(x_1,\ldots,x_n\right) := \left(\sum_{i=1}^n q_i\left(x_i\right)^p\right)^{\frac{1}{p}}</math>
is a semi norm. 
For each <math>p</math> this defines the same topological space.

A straightforward argument involving elementary linear algebra shows that the only finite-dimensional seminormed spaces are those arising as the product space of a normed space and a space with trivial seminorm.  Consequently, many of the more interesting examples and applications of seminormed spaces occur for infinite-dimensional vector spaces.