Normed vector space

A normed vector space is a vector space equipped with a norm, a function assigning each vector a nonnegative size compatible with addition and scalar multiplication. Usually its scalars are real numbers or complex numbers. The norm supplies a distance, making convergence and continuity meaningful alongside the algebraic operations. Normed spaces are basic objects of functional analysis, encompassing both finite-dimensional coordinate spaces and infinite-dimensional spaces of functions or sequences. (ocw.mit.edu)

Definition and basic properties

Let $V$ be a vector space over $\mathbb F=\mathbb R$ or $\mathbb C$. A norm is a function

satisfying, for every $x,y\in V$ and scalar $\alpha$:

1. Definiteness: $\|x\|=0$ if and only if $x=0$.
2. Absolute homogeneity: $\|\alpha x\|=|\alpha|\|x\|$.
3. Triangle inequality: $\|x+y\|\leq\|x\|+\|y\|$.

The pair $(V,\|\cdot\|)$, rather than the underlying vector space alone, specifies the normed space. The axioms imply the reverse triangle inequality,

If definiteness is relaxed so that nonzero vectors may have size zero, the function is a seminorm. Factoring out the subspace of vectors with seminorm zero produces a normed space. (live.ocw.mit.edu)

Metric and topology

Every norm defines a metric by

This metric is translation invariant and satisfies $d(\alpha x,\alpha y)=|\alpha|d(x,y)$. Its open balls

generate the norm topology: a subset is an open set precisely when every point in it lies in a ball contained in that subset. (maths.ox.ac.uk)

A sequence $x_n$ converges to $x$ in norm when $\|x_n-x\|\to0$. Vector addition and scalar multiplication are continuous in this topology. The reverse triangle inequality also shows that the norm itself is a continuous, indeed 1-Lipschitz, real-valued function. These facts ensure that taking limits respects the vector-space operations. (people.math.harvard.edu)

Standard examples

On $\mathbb F^n$, familiar norms include

The second induces Euclidean distance. These norms measure the same vectors differently; in $\mathbb R^2$, their unit balls are respectively a diamond, a disk, and an axis-aligned square. More generally, $(\sum_j|x_j|^p)^{1/p}$ is a norm for $1\leq p<\infty$. (maths.ox.ac.uk)

Important infinite-dimensional examples include:

• Sequence spaces: $\ell^p$ consists of sequences $x=(x_j)$ with $\sum_j|x_j|^p<\infty$, using the analogous $p$-norm. The space $\ell^\infty$ consists of bounded sequences with the supremum norm.
• Continuous functions: $C([a,b])$, the space of scalar-valued continuous functions on a compact interval, has norm $\|f\|_\infty=\max_{t\in[a,b]}|f(t)|$.
• Integrable functions: $L^p$ spaces use $\|f\|_p=(\int|f|^p\,d\mu)^{1/p}$ for $1\leq p<\infty$. Their elements are equivalence classes of functions equal almost everywhere, which makes definiteness hold. (ocw.mit.edu)

Completeness and Banach spaces

A normed space is complete if every Cauchy sequence converges to an element of the space. A complete normed space is a Banach space. Completeness is an additional condition, not a consequence of the norm axioms. The spaces $\ell^p$, $L^p$, and $C([a,b])$ with the norms above are Banach spaces. (ocw.mit.edu)

For an incomplete example, consider finite-support sequences with the $\ell^2$ norm. Truncations of $(2^{-j})_{j\geq1}$ form a Cauchy sequence, but their limit does not have finite support. Every normed space admits a completion: a Banach space containing an isometric copy of it as a dense subspace. This completion is unique up to an isometric linear isomorphism preserving the embedded original space. (ocw.mit.edu)

Equivalent norms and dimension

Two norms on $V$ are equivalent if constants $c,C>0$ exist such that

Equivalent norms induce the same topology, convergent sequences, and Cauchy sequences, and hence preserve completeness. Every pair of norms on a space of finite dimension is equivalent, and every finite-dimensional normed space is complete. (people.math.harvard.edu)

These properties fail in general in infinite dimensions. On finite-support sequences, $\|\cdot\|_1$ and $\|\cdot\|_\infty$ are not equivalent: a vector with $n$ entries equal to one has respective norms $n$ and $1$. Another dimensional distinction concerns compactness: the closed unit ball of a normed space is compact in the norm topology if and only if the space is finite-dimensional. (maths.ox.ac.uk)

Linear operators, subspaces, and quotients

A linear map $T:V\to W$ between normed spaces is continuous exactly when it is bounded: some $M\geq0$ satisfies

Its operator norm is

The space of bounded linear maps is Banach whenever $W$ is Banach. In particular, the continuous dual space, consisting of bounded linear functionals $V\to\mathbb F$, is always Banach, even when $V$ is incomplete. (ocw.mit.edu)

Every linear subspace inherits the ambient norm. A subspace of a Banach space is complete exactly when it is closed. For a closed subspace $Y\subseteq V$, the quotient space carries the norm

If $V$ is Banach, this quotient is also Banach. (live.ocw.mit.edu)

Relation to inner-product spaces

An inner product induces the norm $\|x\|=\sqrt{\langle x,x\rangle}$, but not every norm comes from an inner product. The precise criterion is the parallelogram identity:

When it holds, polarization recovers the unique inducing inner product. A complete inner-product space is a Hilbert space, and therefore a Banach space with this additional geometric structure. (maths.ox.ac.uk)