Fréchet space

A topological vector space X is a Fréchet space if and only if it satisfies the following three properties:

1. It is locally convex.[note 2]
2. Its topology can be induced by a translation-invariant metric, i.e. a metric d: X × X → R such that d(x, y) = d(x+a, y+a) for all a, x, y ∈ X. This means that a subset U of X is open if and only if for every u ∈ U there exists an ε > 0 such that { v : d(v, u) < ε } is a subset of U.
3. Some (or equivalently, every) translation-invariant metric on X inducing the topology of X is complete.
   • Assuming that the other two conditions are satisfied, this condition is equivalent to X being a complete topological vector space, meaning that X is a complete uniform space when it is endowed with its canonical uniformity (this canonical uniformity is independent of any metric on X and is defined entirely in terms of vector subtraction and X 's neighborhoods of the origin; moreover, the uniformity induced by any (topology-defining) translation invariant metric on X is identical to this canonical uniformity).

Note there is no natural notion of distance between two points of a Fréchet space: many different translation-invariant metrics may induce the same topology.

The alternative and somewhat more practical definition is the following: a topological vector space X is a Fréchet space if and only if it satisfies the following three properties:

1. It is a Hausdorff space,
2. Its topology may be induced by a countable family of semi-norms ${\displaystyle ||\cdot ||_{k}}$ k = 0, 1, 2, ... This means that a subset ${\displaystyle U\subset X}$ is open if and only if for every ${\displaystyle u\in U}$ there exists ${\displaystyle K\geq 0}$ and ${\displaystyle \varepsilon >0}$ such that ${\displaystyle \{v\in X:||v-u||_{k}<\varepsilon {\text{ for all }}k\leq K\}}$ is a subset of ${\displaystyle U,}$
3. it is complete with respect to the family of semi-norms.

A family ${\displaystyle {\mathcal {P}}}$ of seminorms on ${\displaystyle X}$ yields a Hausdorff topology if and only if[2]

${\displaystyle \bigcap _{\|\,\cdot \,\|\in {\mathcal {P}}}\{x\in X:\|x\|=0\}=\{0\}.}$

A sequence (x_n) in X converges to x in the Fréchet space defined by a family of semi-norms if and only if it converges to x with respect to each of the given semi-norms.

In contrast to Banach spaces, the complete translation-invariant metric need not arise from a norm. The topology of a Fréchet space does, however, arise from both a total paranorm and an F-norm (the F stands for Fréchet).

Even though the topological structure of Fréchet spaces is more complicated than that of Banach spaces due to the potential lack of a norm, many important results in functional analysis, like the open mapping theorem, the closed graph theorem, and the Banach–Steinhaus theorem, still hold.

• Every Banach space is a Fréchet space, as the norm induces a translation-invariant metric and the space is complete with respect to this metric.
• The space ℝ^ω of all real valued sequences becomes a Fréchet space if we define the k-th semi-norm of a sequence to be the absolute value of the k-th element of the sequence. Convergence in this Fréchet space is equivalent to element-wise convergence.

• The vector space ${\displaystyle C^{\infty }([0,1])}$ of all infinitely differentiable functions ${\displaystyle f:[0,1]\to \mathbb {R} }$ becomes a Fréchet space with the seminorms

  ${\displaystyle \|f\|_{k}=\sup\{|f^{(k)}(x)|:x\in [0,1]\}}$

  for every non-negative integer k. Here, ƒ^(k) denotes the k-th derivative of ƒ, and ƒ⁽⁰⁾ = ƒ. In this Fréchet space, a sequence ${\displaystyle \left(f_{n}\right)\to f}$ of functions converges towards the element ${\displaystyle f\in C^{\infty }([0,1])}$ if and only if for every non-negative integer ${\displaystyle k\geq 0,}$ the sequence ${\displaystyle \left(f_{n}^{(k)}\right)\to f^{(k)}}$ converges uniformly.
• The vector space ${\displaystyle C^{\infty }(\mathbb {R} )}$ of all infinitely differentiable functions ${\displaystyle f:\mathbb {R} \to \mathbb {R} }$ becomes a Fréchet space with the seminorms

  ${\displaystyle \|f\|_{k,n}=\sup\{|f^{(k)}(x)|:x\in [-n,n]\}}$

  for all integers ${\displaystyle k,n\geq 0.}$ Then, a sequence of functions ${\displaystyle (f_{i})\to f}$ converges if and only if for every ${\displaystyle k,n\geq 0,}$ the sequences ${\displaystyle (f_{i}^{(k)})\to f^{(k)}}$ converge compactly.
• The vector space ${\displaystyle C^{m}(\mathbb {R} )}$ of all ${\displaystyle m}$-times continuously differentiable functions ${\displaystyle f:\mathbb {R} \to \mathbb {R} }$ becomes a Fréchet space with the seminorms

  ${\displaystyle \|f\|_{k,n}=\sup\{|f^{(k)}(x)|:x\in [-n,n]\}}$

  for all integers n ≥ 0 and k = 0, ..., m.
• If M is a compact C^∞-manifold and B is a Banach space, then the set C^∞(M, B) of all infinitely-often differentiable functions ƒ : M → B can be turned into a Fréchet space by using as seminorms the suprema of the norms of all partial derivatives. If M is a (not necessarily compact) C^∞-manifold which admits a countable sequence K_n of compact subsets, so that every compact subset of M is contained in at least one K_n, then the spaces C^m(M, B) and C^∞(M, B) are also Fréchet space in a natural manner. As a special case, every smooth finite-dimensional complete manifold M can be made into such a nested union of compact subsets: equip it with a Riemannian metric g which induces a metric d(x, y), choose x ∈ M, and let

  ${\displaystyle K_{n}=\{y\in M|d(x,y)\leq n\}\ .}$

  Let X be a compact C^∞-manifold and V a vector bundle over X. Let C^∞(X, V) denote the space of smooth sections of V over X. Choose Riemannian metrics and connections, which are guaranteed to exist, on the bundles TX and V. If s is a section, denote its j^th covariant derivative by D^js. Then

  ${\displaystyle \|s\|_{n}=\sum _{j=0}^{n}\sup _{x\in M}|D^{j}s|}$

  (where |⋅| is the norm induced by the Riemannian metric) is a family of seminorms making C^∞(M, V) into a Fréchet space.

• Let H be the space of entire (everywhere holomorphic) functions on the complex plane. Then the family of seminorms

  ${\displaystyle \|f\|_{n}=\sup\{|f(z)|:|z|\leq n\}}$

  makes H into a Fréchet space.
• Let 'H be the space of entire (everywhere holomorphic) functions of exponential type τ. Then the family of seminorms

  ${\displaystyle \|f\|_{n}=\sup _{z\in \mathbb {C} }\exp \left[-\left(\tau +{\frac {1}{n}}\right)|z|\right]|f(z)|}$

  makes H into a Fréchet space.

Not all vector spaces with complete translation-invariant metrics are Fréchet spaces. An example is the space L^p([0, 1]) with p < 1. Although this space fails to be locally convex, it is an F-space.

Note that the homeomorphism described in the Anderson–Kadec theorem is not necessarily linear.