Continuous embedding

Let X and Y be two normed vector spaces, with norms ||·||_X and ||·||_Y respectively, such that X ⊆ Y. If the inclusion map (identity function)

${\displaystyle i:X\hookrightarrow Y:x\mapsto x}$

is continuous, i.e. if there exists a constant C > 0 such that

${\displaystyle \|x\|_{Y}\leq C\|x\|_{X}}$

for every x in X, then X is said to be continuously embedded in Y. Some authors use the hooked arrow "↪" to denote a continuous embedding, i.e. "X ↪ Y" means "X and Y are normed spaces with X continuously embedded in Y". This is a consistent use of notation from the point of view of the category of topological vector spaces, in which the morphisms ("arrows") are the continuous linear maps.

• A finite-dimensional example of a continuous embedding is given by a natural embedding of the real line X = R into the plane Y = R², where both spaces are given the Euclidean norm:

${\displaystyle i:\mathbf {R} \to \mathbf {R} ^{2}:x\mapsto (x,0)}$

In this case, ||x||_X = ||x||_Y for every real number X. Clearly, the optimal choice of constant C is C = 1.

• An infinite-dimensional example of a continuous embedding is given by the Rellich–Kondrachov theorem: let Ω ⊆ Rⁿ be an open, bounded, Lipschitz domain, and let 1 ≤ p < n. Set

${\displaystyle p^{*}={\frac {np}{n-p}}.}$

Then the Sobolev space W^1,p(Ω; R) is continuously embedded in the L^p space L^p∗(Ω; R). In fact, for 1 ≤ q < p^∗, this embedding is compact. The optimal constant C will depend upon the geometry of the domain Ω.

• Infinite-dimensional spaces also offer examples of discontinuous embeddings. For example, consider

${\displaystyle X=Y=C^{0}([0,1];\mathbf {R} ),}$

the space of continuous real-valued functions defined on the unit interval, but equip X with the L¹ norm and Y with the supremum norm. For n ∈ N, let f_n be the continuous, piecewise linear function given by

${\displaystyle f_{n}(x)={\begin{cases}-n^{2}x+n,&0\leq x\leq {\tfrac {1}{n}};\\0,&{\text{otherwise.}}\end{cases}}}$

Then, for every n, ||f_n||_Y = ||f_n||_∞ = n, but

${\displaystyle \|f_{n}\|_{L^{1}}=\int _{0}^{1}|f_{n}(x)|\,\mathrm {d} x={\frac {1}{2}}.}$

Hence, no constant C can be found such that ||f_n||_Y ≤ C||f_n||_X, and so the embedding of X into Y is discontinuous.

• Compact embedding