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Split-Isa-AF-MMA

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If $f$ is a linear map from $\mathbb{R}^n$ to $\mathbb{R}^m$, then the dot product of $x$ and the adjoint of $f$ applied to $y$ is equal to the dot product of $f(x)$ and $y$.
If $f$ is a linear map from $\mathbb{R}^n$ to $\mathbb{R}^m$, then the adjoint of $f$ satisfies the following two properties: $x \cdot f^*(y) = f(x) \cdot y$ $f^*(y) \cdot x = y \cdot f(x)$
If $f$ is a linear map from $\mathbb{R}^n$ to $\mathbb{R}^m$, then its adjoint is also a linear map.
The adjoint of the adjoint of a linear map is the original linear map.
The standard basis vectors are linearly independent.
The span of the basis of a vector space is the entire vector space.
Every vector in $\mathbb{R}^n$ is in the span of the standard basis.
If $f$ is a linear function from a Euclidean space to a normed vector space, then there exists a constant $B$ such that for all $x$, we have $||f(x)|| \leq B ||x||$.
A function $f$ is linear if and only if it is bounded linear.
The linearity of a linear operator is equivalent to its bounded linearity.
If $f$ is a bounded linear injection, then $f^{-1}$ is a bounded linear map.
If $f$ is a linear map from a Euclidean space to a normed vector space, then there exists a constant $B$ such that for all $x$, we have $\|f(x)\| \leq B \|x\|$.
If $f$ is a linear map from a normed vector space to a Euclidean space, and $g$ is a linear map from the Euclidean space to the normed vector space such that $g \circ f = id$, then there exists a constant $B > 0$ such that for all $x$, we have $B \cdot \|x\| \leq \|f(x)\|$.
If $f$ is a linear injective map from a normed vector space to a Euclidean space, then there exists a constant $B > 0$ such that $B \|x\| \leq \|f(x)\|$ for all $x$.
If $f$ is a linear map from a Euclidean space to a normed vector space, then $f$ is bounded.
If $h$ is a bilinear function, then there exists a constant $B$ such that for all $x$ and $y$, we have $|h(x,y)| \leq B|x| |y|$.
A function $h$ is bilinear if and only if it is bounded bilinear.
If $h$ is a bilinear function, then there exists a constant $B > 0$ such that for all $x$ and $y$, we have $|h(x, y)| \leq B \cdot |x| \cdot |y|$.
If $f$ is a bounded linear operator, then $f$ is differentiable and its derivative is itself.
If $f$ is a linear function, then $f$ is differentiable and its derivative is itself.
If $f$ is a bounded linear operator, then $f$ is differentiable.
If $f$ is a linear function, then $f$ is differentiable.
If a set of vectors is linearly independent, then it is finite and has at most as many elements as the dimension of the vector space.
If a set of vectors is linearly independent, then it is finite.
If $S$ is a set of linearly independent vectors in $\mathbb{R}^n$, then $|S| \leq n$.
If $S$ is a finite set of vectors in $\mathbb{R}^n$ with more than $n$ elements, then $S$ is linearly dependent.
The projection of a vector $x$ onto a vector $b$ is orthogonal to $x$.
If $x$ is orthogonal to every element of a pairwise orthogonal set $S$, then $x$ is orthogonal to every element of $S \cup \{x\}$.
If $B$ is a finite set of vectors, then there exists a finite set $C$ of vectors such that $C$ is a subset of $B$, $C$ spans the same space as $B$, and the vectors in $C$ are pairwise orthogonal.
For any set $V$ of vectors in Euclidean space, there exists a set $B$ of orthogonal vectors such that $B$ is linearly independent, $B$ spans $V$, $V$ is contained in the span of $B$, and $B$ has the same number of vectors as $V$.
If the span of a set $S$ is not the entire space, then there exists a nonzero vector $a$ such that $a \cdot x = 0$ for all $x \in S$.
If the span of a set $S$ is not the whole space, then there exists a nonzero vector $a$ such that the span of $S$ is contained in the hyperplane $\{x \in \mathbb{R}^n \mid a \cdot x = 0\}$.
If $S$ is a subset of $\mathbb{R}^n$ with dimension less than $n$, then there exists a nonzero vector $a$ such that $S$ is contained in the hyperplane $\{x \in \mathbb{R}^n : a \cdot x = 0\}$.
If two linear functions agree on the standard basis, then they are equal.
If $f$ and $g$ are bilinear functions, and $f$ and $g$ agree on $B \times C$, then $f$ and $g$ agree on $S \times T$ for any $S \subseteq \text{span}(B)$ and $T \subseteq \text{span}(C)$.
If two bilinear functions agree on the standard basis, then they are equal.
The set of absolute values of the inner products of a vector $x$ with the basis vectors is equal to the image of the basis vectors under the function that maps a basis vector to its absolute inner product with $x$.
The infnorm of a vector $x$ is equal to the maximum of the absolute values of the dot products of $x$ with the basis vectors.
The set of absolute values of the inner products of a vector $x$ with the basis vectors is finite and nonempty.
The infnorm of a vector is always nonnegative.
The infnorm of a sum of two vectors is less than or equal to the sum of the infnorms of the vectors.
The infnorm of a vector is zero if and only if the vector is zero.
The infnorm of the zero vector is zero.
The infnorm of a negative number is the same as the infnorm of the positive number.
The infnorm of a difference is the same as the infnorm of the difference with the order of the terms reversed.
The absolute value of the difference of the infnorms of two vectors is less than or equal to the infnorm of the difference of the vectors.
The absolute value of the infnorm of a real number is equal to the infnorm of the real number.
For any vector $x$ in an Euclidean space, the absolute value of the inner product of $x$ with any basis vector is less than or equal to the infnorm of $x$.
The infnorm of a scalar multiple of a vector is equal to the absolute value of the scalar times the infnorm of the vector.
The infinity norm of a scalar multiple of a vector is less than or equal to the absolute value of the scalar times the infinity norm of the vector.
The infnorm of a nonzero vector is positive.
The infnorm of a vector is less than or equal to its norm.
For any vector $x$ in $\mathbb{R}^n$, the Euclidean norm of $x$ is less than or equal to the square root of $n$ times the infinity norm of $x$.
If $f$ converges to $a$, then $\|f\|_\infty$ converges to $\|a\|_\infty$.
$x \cdot y = \|x\| \|y\|$ if and only if $\|x\| y = \|y\| x$.
$\langle x, y \rangle = \|x\| \|y\|$ if and only if $x = \|x\| \frac{y}{\|y\|}$ or $x = -\|x\| \frac{y}{\|y\|}$.
For any two vectors $x$ and $y$, we have $||x + y|| = ||x|| + ||y||$ if and only if $||x||y = ||y||x$.
A set of points is collinear if and only if there exist two points $u$ and $v$ such that every point in the set can be written as $u + cv$ for some scalar $c$.
A set of points in a real vector space is collinear if and only if there exists a nonzero vector $u$ such that for all $x, y \in S$, there exists a scalar $c$ such that $x - y = cu$.
If $T$ is a set of collinear points and $S$ is a subset of $T$, then $S$ is a set of collinear points.
The empty set is collinear.
A single point is collinear.
Two points are collinear.
The points $0$, $x$, and $y$ are collinear if and only if $x = 0$ or $y = 0$ or $y = cx$ for some $c$.
The Cauchy-Schwarz inequality is an equality if and only if the vectors are collinear.
The set of all vectors $x$ such that $a \cdot x = 0$ is a subspace.
The set of all vectors orthogonal to a fixed vector $a$ is a subspace.
The set of points in $\mathbb{R}^n$ that are orthogonal to a given basis vector $k$ is the span of the remaining basis vectors.
If $k$ is a basis vector, then the dimension of the hyperplane $\{x \in \mathbb{R}^n \mid k \cdot x = 0\}$ is $n-1$.
The dimension of the hyperplane $\{x \in \mathbb{R}^n \mid a \cdot x = 0\}$ is $n-1$.
If $S$ is a set of vectors in $\mathbb{R}^n$ such that $\dim(S) = n-1$, then there exists a vector $a$ such that $S$ is the set of vectors $x$ such that $a \cdot x = 0$.
A set $S$ has dimension $n-1$ if and only if there exists a nonzero vector $a$ such that $S$ is the set of all vectors $x$ such that $a \cdot x = 0$.
If a set of vectors is pairwise orthogonal and does not contain the zero vector, then it is linearly independent.
If a set of vectors is pairwise orthogonal, then it is finite.
The set of all vectors orthogonal to a given vector $x$ is a subspace.
The set of all vectors orthogonal to a set of vectors is a subspace.
If $a$ is in the span of $S$ and $x$ is orthogonal to every element of $S$, then $x$ is orthogonal to $a$.
If $S$ is a set of pairwise orthogonal vectors in a Euclidean space, and $x$ is a vector in the span of $S$, then $x$ is orthogonal to $a - \sum_{b \in S} \frac{b \cdot a}{b \cdot b} b$.
If $S$ is a finite set of pairwise orthogonal vectors, then there exists a finite set $U$ of pairwise orthogonal vectors such that $S \cup U$ spans the same space as $S \cup T$.
If $S$ is a set of pairwise orthogonal vectors, then there exists a set $U$ of pairwise orthogonal vectors such that $S \cup U$ is pairwise orthogonal and $\text{span}(S \cup U) = \text{span}(S \cup T)$.
If $S$ is a set of pairwise orthogonal vectors, then there exists a set $U$ of pairwise orthogonal vectors such that $U \cap (S \cup \{0\}) = \emptyset$ and $\text{span}(S \cup U) = \text{span}(S \cup T)$.
Every subspace of a Euclidean space has an orthogonal basis.
If $S$ is a subspace of a Euclidean space, then there exists a basis $B$ of $S$ consisting of orthogonal vectors.
Every subspace of a Euclidean space has an orthonormal basis.
If $S$ is a subset of $T$, then there exists a nonzero vector $x$ in $T$ that is orthogonal to every vector in $S$.
If $S$ is a subspace of $\mathbb{R}^n$ with dimension less than $n$, then there exists a nonzero vector $x$ that is orthogonal to every vector in $S$.
If $x$ is a vector in $\mathbb{R}^n$ with $n \geq 2$, then there exists a nonzero vector $y$ that is orthogonal to $x$.
Given a vector $x$ and a subspace $S$, there exist vectors $y$ and $z$ such that $y \in S$, $z$ is orthogonal to $S$, and $x = y + z$.
If $x + y = x' + y'$ and $x, x' \in \operatorname{span}(S)$ and $y, y' \in \operatorname{span}(T)$ and $a \perp b$ for all $a \in S$ and $b \in T$, then $x = x'$ and $y = y'$.
For any vector $a$, there exists a set of vectors $S$ such that $a \in S$, $S$ is pairwise orthogonal, and $S$ spans the entire space.
For any nonzero vector $a$, there exists a finite set of orthogonal vectors $S$ such that $a \in S$, $0 \notin S$, $S$ is independent, and $S$ spans the entire space.
If $a$ is a unit vector, then there exists an orthonormal basis $S$ of $\mathbb{R}^n$ such that $a \in S$.
If $A$ and $B$ are orthogonal, then the dimension of their union is the sum of their dimensions.
If $A$ is a subspace of $B$, then the dimension of the subspace of $B$ orthogonal to $A$ is equal to the dimension of $B$ minus the dimension of $A$.
If $f$ is a linear function, then $f(x) \to 0$ as $x \to 0$.
If $f$ is a linear map, then $f$ is continuous at every point.
If $f$ is a linear function, then $f$ is continuous at every point $x$ in every set $S$.
If $f$ is a linear map, then $f$ is continuous.
If $f$ converges to $l$ and $h$ is linear, then $h(f)$ converges to $h(l)$.
If $f$ is a continuous function from a topological space $X$ to a topological space $Y$ and $g$ is a linear function from $Y$ to a normed vector space $Z$, then the composition $g \circ f$ is continuous.