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Definition

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)

i : X ↪ Y : x ↦ x {\displaystyle i:X\hookrightarrow Y:x\mapsto x}

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

‖ x ‖ Y ≤ C ‖ x ‖ X {\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.

Examples

• 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:

i : R → R 2 : x ↦ ( x , 0 ) {\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

p ∗ = n p n − p . {\displaystyle p^{*}={\frac {np}{n-p}}.} Then the Sobolev space W1,p(Ω; R) is continuously embedded in the Lp space Lp∗(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

X = Y = C 0 ( [ 0 , 1 ] ; R ) , {\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 fn be the continuous, piecewise linear function given by f n ( x ) = { − n 2 x + n , 0 ≤ x ≤ 1 n ; 0 , otherwise. {\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, ||fn||Y = ||fn||∞ = n, but ‖ f n ‖ L 1 = ∫ 0 1 | f n ( x ) | d x = 1 2 . {\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 ||fn||Y ≤ C||fn||X, and so the embedding of X into Y is discontinuous.

See also

• Compact embedding

• Renardy, M. & Rogers, R.C. (1992). An Introduction to Partial Differential Equations. Springer-Verlag, Berlin. ISBN 3-540-97952-2.