Distances are often more important than coordinates. A rotation of the plane changes every coordinate pair, but it does not change the shape of any figure; translating a function in a Hilbert space may move it to a different location, but the norm can remain unchanged. An isometry is the abstraction of this exact preservation. It is the map that lets geometry be transported without distortion, whether the ambient object is a [metric space](/page/Metric%20Space), an [inner product space](/page/Inner%20Product%20Space), a [Hilbert space](/page/Hilbert%20Space), or a [Riemannian manifold](/page/Cambridge%20III%20Riemannian%20Geometry). The word has several closely related uses. In metric geometry it means preservation of distance. In linear analysis it often means preservation of norm, especially for bounded operators on normed or Hilbert spaces. In finite-dimensional [inner product](/page/Inner%20Product) geometry it means preservation of the inner product itself, giving the familiar orthogonal and unitary maps. In Riemannian geometry it means preservation of the metric tensor through a diffeomorphism. These are not competing ideas: they are manifestations of the same principle at different levels of structure. ## Definition The most flexible setting is a metric space, where the only primitive geometric data is the distance between two points. A map deserves to be called rigid in this setting if every pairwise distance is preserved, since all metric statements about balls, convergence, diameter, and Cauchy behaviour are encoded by those distances. [definition: Isometry] Let $(X,d_X)$ and $(Y,d_Y)$ be metric spaces. A map $f:X\to Y$ is an isometry if \begin{align*} d_Y(f(x),f(y)) = d_X(x,y) \end{align*} for all $x,y\in X$. [/definition] This definition does not require $f$ to be onto. Many authors reserve the word isometry for the surjective case and call the general case an isometric embedding. When a page needs to stress that the source sits inside the target as an undistorted copy, the embedding terminology makes that role explicit. [definition: Isometric Embedding] Let $(X,d_X)$ and $(Y,d_Y)$ be metric spaces. An isometric embedding is an isometry $f:X\to Y$. [/definition] A map that preserves all distances cannot identify two distinct points, since the distance between distinct points is positive. Sometimes, however, the goal is not to place $X$ inside $Y$, but to identify $X$ and $Y$ as the same metric object with different labels. That requires a distance-preserving map that reaches every point of the target. [definition: Surjective Isometry] Let $(X,d_X)$ and $(Y,d_Y)$ be metric spaces. A surjective isometry is an isometry $f:X\to Y$ such that for every $y\in Y$ there exists $x\in X$ with $f(x)=y$. [/definition] The surjective condition is what turns an undistorted copy into a genuine identification of metric spaces. For this identification to be usable, it must work in both directions: after relabelling $X$ as $Y$, we should be able to recover $X$ from $Y$ without introducing distortion. The next result records that no extra hypothesis is needed for this reverse map; surjectivity and distance preservation already force the inverse to preserve distance. [quotetheorem:8698] This result justifies treating surjective isometries as genuine sameness of metric spaces rather than merely as special maps between them. The surjectivity hypothesis is essential: a non-surjective isometric embedding, such as the inclusion of a closed interval into a larger interval, preserves every distance in its domain but has no inverse defined on the whole target. In that example, distances inside the smaller interval are unchanged, but points of the larger interval outside the image have no preimage at all, so statements about the whole target cannot be transported back. The theorem also explains why the inverse map requires no separate continuity or Lipschitz argument. Once every point of the target has a unique preimage, the equation defining distance preservation can be read in reverse, so metric balls, convergent sequences, Cauchy behavior, diameters, and other distance-defined data transfer back to the source. This is the practical reason the surjective condition appears throughout later invariance statements: it lets one replace a metric space by an isometric model without changing what is true about the whole space. Bijectivity alone would not be enough either, since an arbitrary bijection can stretch distances; the point is that the inverse inherits distance preservation from the original map. This is why surjective isometries are the maps that allow metric arguments, examples, and invariants to be transported back and forth without changing their content. The same observation also explains the categorical role of these maps: they are precisely the morphisms that can be undone within the metric category. [remark: Categorical Role of Surjective Isometries] In the category whose objects are metric spaces and whose morphisms are distance-preserving maps, the isomorphisms are precisely the surjective isometries. [/remark] This categorical viewpoint is useful, but many analytic settings still need non-surjective isometries, especially when one space embeds into a larger one. In normed vector spaces, distance is usually generated by a norm. For a linear map, preserving all distances is equivalent to preserving the norm of every vector. This is why the operator-theoretic definition is often written using only $\|Tx\|$. [definition: Linear Isometry] Let $X$ and $Y$ be normed vector spaces over $\mathbb R$ or $\mathbb C$, with norms $\|\cdot\|_X$ and $\|\cdot\|_Y$. A linear map $T:X\to Y$ is a linear isometry if \begin{align*} \|Tx\|_Y = \|x\|_X \end{align*} for all $x\in X$. [/definition] Hilbert spaces are the central normed spaces of analysis because their norms come from inner products. To study shifts, partial symmetries, and unitary dynamics, we need a version of isometry for bounded operators that preserves Hilbert norm while allowing the range to be a proper closed subspace. [definition: Hilbert Space Isometry] Let $H$ and $K$ be Hilbert spaces over $\mathbb R$ or $\mathbb C$, and let $\mathcal{L}(H,K)$ denote the space of bounded linear maps from $H$ to $K$. An operator $V\in \mathcal{L}(H,K)$ is a Hilbert space isometry if \begin{align*} \|Vx\|_K = \|x\|_H \end{align*} for all $x\in H$. [/definition] Finite-dimensional geometry often asks not only whether lengths are preserved, but whether angles and orthogonality are preserved. Since an inner product records both length and angle, the classical finite-dimensional definition is stated directly in terms of inner products. [definition: Inner Product Isometry] Let $V$ be an inner product space over $\mathbb R$ or $\mathbb C$, with inner product $(\cdot,\cdot)_V$. An endomorphism $A:V\to V$ is an inner product isometry if \begin{align*} (Av,Aw)_V = (v,w)_V \end{align*} for all $v,w\in V$. [/definition] A curved space has a different inner product at each tangent space, and a smooth map compares those inner products through its differential. The Riemannian definition is needed to say that lengths, angles, and the metric tensor are transported without distortion at every point of the manifold. [definition: Riemannian Isometry] Let $(M,g)$ and $(N,h)$ be Riemannian manifolds. A smooth map $f:M\to N$ is a Riemannian isometry if $f$ is a diffeomorphism and \begin{align*} h_{f(p)}(df_p(u),df_p(v)) = g_p(u,v) \end{align*} for every $p\in M$ and every $u,v\in T_pM$. [/definition] Here $T_pM$ denotes the tangent space to $M$ at $p$, and $df_p:T_pM\to T_{f(p)}N$ is the derivative, or pushforward, of $f$ at that point. The symbols $g_p$ and $h_{f(p)}$ mean that the Riemannian metrics $g$ and $h$ are being evaluated on tangent vectors at the points $p$ and $f(p)$ respectively. Equivalently, in the language of pullbacks, the condition is $f^*h=g$, where $f^*h$ is the metric on $M$ obtained by measuring pushed-forward tangent vectors with $h$. In this page, a Riemannian isometry means a diffeomorphism satisfying this metric-preservation condition. If a later quoted result describes a Riemannian isometry by saying that it is smooth and preserves the metric on tangent vectors, that wording should be read in this diffeomorphic setting: smoothness and metric preservation are the local hypotheses being used to control lengths, not a replacement of the global bijective requirement. This compact tensor equation is useful because it behaves well under composition and local coordinate calculations. ## Equivalent Characterisations The metric definition asks for every pairwise distance to be preserved. In normed vector spaces, a linear map can be checked at the origin because the vector-space structure turns every distance into a norm of a difference. [quotetheorem:8699] This result explains why functional analysts often write the definition of a linear isometry using only norms. The distance statement is still present, but it has been compressed by linearity. For Hilbert spaces, the next natural question is whether preserving norm also preserves the inner product structure used in orthogonality and projection arguments. [quotetheorem:8700] The theorem is often the most useful test in practice. A calculation involving inner products can verify an isometry without separately checking every norm. In finite dimensions, this inner-product test becomes a matrix equation once an [orthonormal basis](/page/Orthonormal%20Basis) has been chosen. For that formulation, $\mathrm{End}(V)$ denotes the algebra of linear maps from $V$ to itself, $\alpha^*$ denotes the adjoint of the linear map $\alpha$, characterized by $\langle \alpha v,w\rangle=\langle v,\alpha^*w\rangle$, and $A^\dagger$ denotes the conjugate transpose, or adjoint, of a complex matrix $A$. [quotetheorem:439] The real condition says that the columns of the matrix $A$ form an orthonormal basis. The complex condition is the analogous statement using the adjoint matrix. This finite-dimensional characterisation matters because it turns the geometric requirement of preserving every length and angle into a finite algebraic check: multiply the matrix by its transpose or adjoint and compare with the identity. It is the form used in computations with rotations, reflections, changes of orthonormal basis, and unitary evolution. Its limitation is also important: it depends on choosing orthonormal bases and on linearity, so it does not by itself describe translations, curved-space isometries, or nonlinear maps between metric spaces. The examples below separate these cases by showing which isometries are linear matrix examples and which preserve distance for different reasons. ## Standard Examples The basic Euclidean isometries show why the concept is geometric rather than merely algebraic. Translations are not linear maps, but they preserve every distance; rotations and reflections are linear and preserve inner products. [example: Euclidean Translations and Orthogonal Maps] Let $a\in\mathbb R^n$, and define $T_a:\mathbb R^n\to\mathbb R^n$ by $T_a(x)=x+a$. For $x,y\in\mathbb R^n$, \begin{align*} T_a(x)-T_a(y)=(x+a)-(y+a)=x-y. \end{align*} Taking Euclidean norms gives \begin{align*} |T_a(x)-T_a(y)|=|x-y|. \end{align*} The map is onto because, for any $z\in\mathbb R^n$, choosing $x=z-a$ gives \begin{align*} T_a(z-a)=(z-a)+a=z. \end{align*} Thus $T_a$ is a surjective isometry. Now let $Q\in\mathbb R^{n\times n}$ satisfy $Q^\top Q=I$, and define $F(x)=Qx$. For $x,y\in\mathbb R^n$, put $v=x-y$. Then \begin{align*} F(x)-F(y)=Qx-Qy=Q(x-y)=Qv. \end{align*} Using the Euclidean inner product, \begin{align*} |Qv|^2=(Qv,Qv)_{\mathbb R^n}=(Qv)^\top(Qv)=v^\top Q^\top Qv. \end{align*} Since $Q^\top Q=I$, \begin{align*} v^\top Q^\top Qv=v^\top Iv=v^\top v=(v,v)_{\mathbb R^n}=|v|^2. \end{align*} Both $|Qv|$ and $|v|$ are nonnegative, so $|Qv|^2=|v|^2$ implies $|Qv|=|v|$. Therefore \begin{align*} |F(x)-F(y)|=|x-y|. \end{align*} Finally, $\det(Q^\top Q)=\det(I)=1$, while $\det(Q^\top Q)=\det(Q^\top)\det(Q)=(\det Q)^2$, so $\det Q\ne 0$. Hence $Q$ is invertible, and for every $z\in\mathbb R^n$ the vector $x=Q^{-1}z$ satisfies $F(x)=z$. Thus $x\mapsto Qx$ is also a surjective isometry. [/example] This example separates two phenomena that are sometimes conflated. An isometry between metric spaces need not be linear, while a linear isometry is an isometry of the induced metric spaces only after the metric has been induced by a norm. Infinite-dimensional Hilbert spaces introduce a new boundary: an isometry need not be onto. This is one of the reasons the terms isometry and unitary operator should not be treated as interchangeable. [example: The Unilateral Shift on Sequence Space] Let $x=(x_1,x_2,x_3,\ldots)\in \ell^2$, and define \begin{align*} Sx=(0,x_1,x_2,x_3,\ldots). \end{align*} The shifted sequence is still square-summable because \begin{align*} \sum_{n=1}^{\infty}|(Sx)_n|^2=|0|^2+\sum_{n=2}^{\infty}|x_{n-1}|^2=\sum_{m=1}^{\infty}|x_m|^2<\infty. \end{align*} Thus $Sx\in\ell^2$. The map $S$ is linear: for $\alpha,\beta\in\mathbb C$ and $x,y\in\ell^2$, \begin{align*} S(\alpha x+\beta y)=(0,\alpha x_1+\beta y_1,\alpha x_2+\beta y_2,\ldots)=\alpha Sx+\beta Sy. \end{align*} Its norm is preserved because \begin{align*} \|Sx\|_{\ell^2}^2=\sum_{n=1}^{\infty}|(Sx)_n|^2=\sum_{m=1}^{\infty}|x_m|^2=\|x\|_{\ell^2}^2. \end{align*} Since both norms are nonnegative, $\|Sx\|_{\ell^2}=\|x\|_{\ell^2}$ for every $x\in\ell^2$. Hence $S$ is a Hilbert space isometry. It is not surjective. The vector $e_1=(1,0,0,\ldots)$ belongs to $\ell^2$, since \begin{align*} \|e_1\|_{\ell^2}^2=1^2+0^2+0^2+\cdots=1. \end{align*} But every vector of the form $Sx$ has first coordinate $0$, so $Sx=e_1$ would force $0=1$ in the first coordinate. Therefore $e_1\notin S(\ell^2)$, and the unilateral shift is an isometry whose range is a proper subspace of $\ell^2$. [/example] The unilateral shift is a central example in operator theory. It preserves all norms and inner products on its domain, but it is not a unitary operator because it fails to be onto. A simple non-example shows why exact preservation matters. Many maps preserve angles, ratios, or topology, but those weaker forms of structure are not isometry. [example: Dilation Is Not an Isometry] Let $\lambda\in\mathbb R$ with $\lambda>0$ and $\lambda\ne 1$, and define $D_\lambda:\mathbb R^n\to\mathbb R^n$ by $D_\lambda(x)=\lambda x$. For $x,y\in\mathbb R^n$, \begin{align*} D_\lambda(x)-D_\lambda(y)=\lambda x-\lambda y=\lambda(x-y). \end{align*} Using homogeneity of the Euclidean norm and $\lambda>0$, \begin{align*} |D_\lambda(x)-D_\lambda(y)|=|\lambda(x-y)|=\lambda |x-y|. \end{align*} If $x\ne y$, then $|x-y|>0$. Since $\lambda\ne 1$, \begin{align*} \lambda |x-y|-|x-y|=(\lambda-1)|x-y|\ne 0. \end{align*} Thus $|D_\lambda(x)-D_\lambda(y)|\ne |x-y|$, so $D_\lambda$ is not an isometry. The map is nevertheless a homeomorphism: its inverse is $D_{1/\lambda}$, because \begin{align*} D_{1/\lambda}(D_\lambda(x))=\frac{1}{\lambda}(\lambda x)=x \end{align*} and \begin{align*} D_\lambda(D_{1/\lambda}(x))=\lambda\left(\frac{1}{\lambda}x\right)=x. \end{align*} It also sends the line segment from $x$ to $y$ to the line segment from $\lambda x$ to $\lambda y$, since for $t\in[0,1]$, \begin{align*} D_\lambda((1-t)x+ty)=\lambda((1-t)x+ty)=(1-t)\lambda x+t\lambda y. \end{align*} So dilation preserves this affine shape, but it changes every nonzero distance by the factor $\lambda$, which is why it is not an isometry. [/example] This failure is useful because it marks the line between geometric similarity and geometric congruence. Isometries preserve scale, not only shape. Riemannian examples show that the definition is local in tangent spaces but global through diffeomorphism. The circle already displays this structure without the complexity of curvature. [example: Rotations of the Circle] Let $S^1\subset\mathbb R^2$ have the Riemannian metric induced by the Euclidean inner product. For a fixed angle $\theta\in\mathbb R$, define $R_\theta:S^1\to S^1$ by \begin{align*} R_\theta(\cos t,\sin t)=(\cos(t+\theta),\sin(t+\theta)). \end{align*} Equivalently, for $(u,v)\in S^1$, \begin{align*} R_\theta(u,v)=(\cos\theta\,u-\sin\theta\,v,\sin\theta\,u+\cos\theta\,v). \end{align*} This formula sends $S^1$ to itself, because if $u^2+v^2=1$, then \begin{align*} (\cos\theta\,u-\sin\theta\,v)^2+(\sin\theta\,u+\cos\theta\,v)^2=(\cos^2\theta+\sin^2\theta)u^2+(\sin^2\theta+\cos^2\theta)v^2+(-2\sin\theta\cos\theta+2\sin\theta\cos\theta)uv=u^2+v^2=1. \end{align*} The inverse map is $R_{-\theta}$. Indeed, using $\cos(-\theta)=\cos\theta$ and $\sin(-\theta)=-\sin\theta$, the first coordinate of $R_{-\theta}(R_\theta(u,v))$ is \begin{align*} \cos\theta(\cos\theta u-\sin\theta v)+\sin\theta(\sin\theta u+\cos\theta v)=(\cos^2\theta+\sin^2\theta)u+(-\cos\theta\sin\theta+\sin\theta\cos\theta)v=u. \end{align*} The second coordinate is \begin{align*} -\sin\theta(\cos\theta u-\sin\theta v)+\cos\theta(\sin\theta u+\cos\theta v)=(-\sin\theta\cos\theta+\cos\theta\sin\theta)u+(\sin^2\theta+\cos^2\theta)v=v. \end{align*} The same calculation with $\theta$ replaced by $-\theta$ gives $R_\theta(R_{-\theta}(u,v))=(u,v)$, so $R_\theta$ is bijective with smooth inverse $R_{-\theta}$. At the point $p(t)=(\cos t,\sin t)$, the unit tangent vector is $p'(t)=(-\sin t,\cos t)$. Since $R_\theta$ is the restriction of a linear rotation of $\mathbb R^2$, its differential sends this vector to \begin{align*} d(R_\theta)_{p(t)}(-\sin t,\cos t)=(-\sin(t+\theta),\cos(t+\theta)). \end{align*} Every tangent vector at $p(t)$ has the form $a(-\sin t,\cos t)$. For two such tangent vectors $a(-\sin t,\cos t)$ and $b(-\sin t,\cos t)$, the induced metric after applying the differential is \begin{align*} \bigl(a(-\sin(t+\theta),\cos(t+\theta)),b(-\sin(t+\theta),\cos(t+\theta))\bigr)_{\mathbb R^2}=ab(\sin^2(t+\theta)+\cos^2(t+\theta))=ab. \end{align*} Before applying the differential, the induced metric is \begin{align*} \bigl(a(-\sin t,\cos t),b(-\sin t,\cos t)\bigr)_{\mathbb R^2}=ab(\sin^2 t+\cos^2 t)=ab. \end{align*} Thus $R_\theta$ preserves the induced Riemannian metric at every point of $S^1$, so it is a Riemannian isometry. [/example] This example is also a model for group actions by isometries: the circle group acts on $S^1$ by rotations, and each group element gives a Riemannian isometry. ## Properties The first structural property is stability under composition. If one map introduces no distortion and the next map introduces no distortion, their composite introduces no distortion. [quotetheorem:8701] Together with identity maps, this theorem explains why surjective isometries form symmetry groups. It also underlies the use of isometries as structure-preserving morphisms. Distance preservation is stronger than continuity. It controls the image of every convergent sequence and every [Cauchy sequence](/page/Cauchy%20Sequence) with no need for separate estimates. [quotetheorem:8702] The next question is whether an undistorted copy also carries the right topology. Continuity of $f$ only controls motion from $X$ into $Y$; it does not by itself guarantee that the topology of $X$ can be recovered from the image $f(X)$. For an isometric embedding, exact distance preservation gives this stronger conclusion: open balls in the source correspond to open balls in the image with the subspace metric, so the topology is transported without loss. [quotetheorem:8703] This theorem says that the induced topology on the image agrees with the original topology on the source. Since Cauchy sequences are defined entirely by distances, the next preservation result concerns completeness. [quotetheorem:8704] Completeness is a metric property in the strongest sense: it depends only on which sequences are Cauchy and where their limits must lie. A surjective isometry preserves Cauchy behavior in both directions, so a complete target cannot hide a missing limit from the source, and a complete source cannot acquire new metric gaps after relabelling. This matters whenever a construction is made after replacing a space by an isometric model: Banach-space arguments, fixed-point principles, and completion procedures are valid only when the replacement has the same limit behavior. Surjectivity is where the invariance statement becomes stronger than the embedding statement. An isometric copy of a complete space may sit as a closed proper subset of a larger complete space, so completeness of the source says nothing about all of the target. Conversely, an incomplete space may embed isometrically into its completion without becoming complete, because the missing limits live outside the image. For example, the rational line embeds isometrically into the real line, but the real limits of rational Cauchy sequences need not be rational. Thus completeness is preserved under metric sameness, not under arbitrary placement inside a larger metric space. This distinction is useful later because operator theory often studies maps that preserve norm but are not onto. Such maps faithfully represent the domain inside a larger Hilbert space, while surjective isometries identify two Hilbert spaces as complete metric objects with the same limiting processes. Completeness is a metric property, but Hilbert-space analysis also needs algebraic criteria for operators. The adjoint is the natural tool for converting the inner-product preservation law into an operator identity, which is why the next characterisation is used constantly in operator theory. [quotetheorem:8705] When $H=K$, surjectivity is equivalent to $VV^*=I_H$ in addition to $V^*V=I_H$. In that case the isometry is a unitary operator. On a Riemannian manifold, the analogous question is how the pointwise metric condition affects lengths of curves. [quotetheorem:8706] This theorem is the bridge from tensor preservation to metric geometry on manifolds. It explains why a Riemannian isometry is also distance-preserving for the induced geodesic distance when the distance is defined by infima of curve lengths. ## Relationship to Other Concepts Isometry is stronger than [continuity](/page/Continuity), stronger than being a homeomorphism when the map is surjective, and different from mere algebraic invertibility. A bijective continuous map may distort distances heavily; an isometry cannot distort any distance at all. In linear algebra, real finite-dimensional isometries are the elements of the [orthogonal group](/page/Orthogonal%20Group), while complex finite-dimensional isometries are the elements of the unitary group. These groups are central examples of [group actions](/page/Group%20Action), since they act on Euclidean or Hermitian space by distance-preserving transformations. In functional analysis, Hilbert space isometries sit between general bounded operators and unitary operators. Every unitary operator is a surjective Hilbert space isometry, but the unilateral shift shows that a Hilbert space isometry need not be unitary. This distinction is important in the study of [bounded linear operators](/page/Bounded%20Linear%20Operator), invariant subspaces, and operator models. In metric geometry, an isometric embedding identifies one space with a subset of another without changing distances. This is stronger than a topological embedding and stronger than a Lipschitz embedding with distortion constants. It is the natural language for completions, universal spaces, and rigidity problems. In Riemannian geometry, the isometry condition $f^*h=g$ says that the metric tensor itself is preserved. Since gradients, geodesics, volume forms, curvature, and Laplace-type operators are built from the metric, Riemannian isometries are the symmetries that preserve the full geometric structure of the manifold. [remark: Isometry Versus Similarity] A similarity may multiply every distance by a fixed positive constant. An isometry is the special case where that constant is $1$. Thus dilations of Euclidean space are similarities, but only the dilation factor $1$ gives an isometry. [/remark] [remark: Isometry Versus Unitary Operator] In Hilbert space terminology, an isometry need not be surjective. A unitary operator is a surjective Hilbert space isometry from a Hilbert space onto itself. [/remark] ## Beyond and Connections Isometries sit at the intersection of metric geometry, linear algebra, functional analysis, and Riemannian geometry. In a metric space, they are the maps that preserve the distance function exactly. In a Hilbert space, linear isometries are also controlled by the inner product, so they preserve orthogonality, projections, and many arguments built from the [Pythagorean theorem](/theorems/3266). In finite dimensions, the same condition becomes the familiar orthogonal or unitary matrix equation. The concept also marks a boundary between rigid and flexible equivalence. An isometry identifies spaces without changing any metric measurement, while weaker notions such as homeomorphism, diffeomorphism, similarity, or quasi-isometry preserve only selected structure. This distinction explains why isometries are central when the actual metric matters, but too restrictive when one only wants topology, smooth structure, large-scale geometry, or approximate shape. On Riemannian manifolds, isometries connect the local and global viewpoints: the differential preserves each tangent-space inner product, while the map itself preserves the induced geometry. This is why isometries interact naturally with geodesics, curvature, symmetry groups, and homogeneous spaces. The same word therefore names one idea across several settings: no metric information is lost, although the tools for checking that fact depend strongly on the structure available. ## References [Metric Space](/page/Metric%20Space). [Hilbert Space](/page/Hilbert%20Space). [Inner Product Space](/page/Inner%20Product%20Space). [Linear Map](/page/Linear%20Map). [Cambridge III Riemannian Geometry](/page/Cambridge%20III%20Riemannian%20Geometry). John B. Conway, *A Course in Functional Analysis* (1990). Manfredo P. do Carmo, *Riemannian Geometry* (1992). Walter Rudin, *Functional Analysis* (1991).