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description/proof of that map preimage of range is whole domain
Topics
About:
set
The table of contents of this article
Starting Context
Target Context
-
The reader will have a description and a proof of the proposition that for any map, the map preimage of the range is the whole domain
Orientation
There is a list of definitions discussed so far in this site.
There is a list of propositions discussed so far in this site.
Main Body
1: Structured Description
Here is the rules of Structured Description.
Entities:
\(S_1\): \(\in \{\text{ the sets }\}\)
\(S_2\): \(\in \{\text{ the sets }\}\)
\(f\): \(S_1 \to S_2\)
//
Statements:
\(f^{-1} (f (S_1)) = S_1\).
//
2: Natural Language Description
For any sets, \(S_1, S_2\), and any map, \(f: S_1 \to S_2\), \(f^{-1} (f (S_1)) = S_1\).
3: Proof
Obviously, \(f^{-1} (f (S_1)) \subseteq S_1\).
By the proposition that for any map, the composition of the preimage after the map of any subset is identical if and only if it is contained in the argument set, \(f^{-1} (f (S_1)) = S_1\).
References
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<The previous article in this series | The table of contents of this series | The next article in this series>
definition of product topological space
Topics
About:
topological space
The table of contents of this article
Starting Context
Target Context
-
The reader will have a definition of product topological space.
Orientation
There is a list of definitions discussed so far in this site.
There is a list of propositions discussed so far in this site.
Main Body
1: Structured Description 1
Here is the rules of Structured Description.
Entities:
\( A\): \(\in \{\text{ the possibly uncountable infinite index sets }\}\)
\( \{T_\alpha \vert \alpha \in A\}\): \(T_\alpha \in \{\text{ the topological spaces }\}\)
\(*T\): \(= \times_{\alpha \in A} T_\alpha\) with the product topology
//
Conditions:
//
2: Natural Language Description 1
For any possibly uncountable infinite index set, \(A\), and any topological spaces, \(\{T_\alpha \vert \alpha \in A\}\), the product set, \(T := \times_{\alpha \in A} T_\alpha\), with the product topology
3: Structured Description 2
Here is the rules of Structured Description.
Entities:
\( \{T_1, ..., T_n\}\): \(T_j \in \{\text{ the topological spaces }\}\)
\(*T\): \(= T_1 \times ... \times T_n\) with the product topology
//
Conditions:
//
4: Natural Language Description 2
For any topological spaces, \(\{T_1, ..., T_n\}\), the product set, \(T := T_1 \times ... \times T_n\), with the product topology
References
<The previous article in this series | The table of contents of this series | The next article in this series>
<The previous article in this series | The table of contents of this series | The next article in this series>
definition of product map
Topics
About:
set
The table of contents of this article
Starting Context
Target Context
-
The reader will have a definition of product map.
Orientation
There is a list of definitions discussed so far in this site.
There is a list of propositions discussed so far in this site.
Main Body
1: Structured Description 1
Here is the rules of Structured Description.
Entities:
\( A\): \(\in \{\text{ the possibly uncountable index sets }\}\)
\( \{S_\alpha\}\): \(\alpha \in A\), \(S_\alpha \in \{\text{ the sets }\}\)
\( \{S'_\alpha\}\): \(\alpha \in A\), \(S'_\alpha \in \{\text{ the sets }\}\)
\( \{f_\alpha\}\): \(\alpha \in A\), \(: S_\alpha \to S'_\alpha\)
\(*\times_{\alpha \in A} f_\alpha\): \(:\times_{\alpha \in A} S_\alpha \to \times_{\alpha \in A} S'_\alpha, (\alpha \mapsto f (\alpha)) \mapsto (\alpha \mapsto f_\alpha (f (\alpha)))\)
//
Conditions:
//
2: Natural Language Description 1
For any possibly uncountable index set, \(A\), any sets, \(\{S_\alpha \vert \alpha \in A\}\), any sets, \(\{S'_\alpha \vert \alpha \in A\}\), and any maps, \(\{f_\alpha: S_\alpha \to S'_\alpha\}\), \(\times_{\alpha \in A} f_\alpha :\times_{\alpha \in A} S_\alpha \to \times_{\alpha \in A} S'_\alpha\), \((\alpha \mapsto f (\alpha)) \mapsto (f': \alpha \mapsto f_\alpha (f (\alpha)))\)
3: Structured Description 2
Here is the rules of Structured Description.
Entities:
\( J\): \(= \{1, ..., n\}\)
\( \{S_j\}\): \(j \in J\), \(S_j \in \{\text{ the sets }\}\)
\( \{S'_j\}\): \(j \in J\), \(S'_j \in \{\text{ the sets }\}\)
\( \{f_j\}\): \(j \in J\), \(: S_j \to S'_j\)
\(*f_1 \times f_2 \times ... \times f_n\): \(: S_1 \times S_2 \times ... \times S_n \to S'_1 \times S'_2 \times ... \times S'_n, (p_1, p_2, ..., p_n) \mapsto (f_1 (p_1), f_2 (p_2), ..., f_n (p_n))\)
Conditions:
//
4: Natural Language Description 2
For any finite number of sets, \(S_1, S_2, ..., S_n\), any same number of sets, \(S'_1, S'_2, ..., S'_n\), and any same number of maps, \(f_1: S_1 \to S'_1, f_2: S_2 \to S'_2, ..., f_n: S_n \to S'_n\), \(f_1 \times f_2 \times ... \times f_n: S_1 \times S_2 \times ... \times S_n \to S'_1 \times S'_2 \times ... \times S'_n\), \((p_1, p_2, ..., p_n) \mapsto (f_1 (p_1), f_2 (p_2), ..., f_n (p_n))\)
References
<The previous article in this series | The table of contents of this series | The next article in this series>
<The previous article in this series | The table of contents of this series | The next article in this series>
definition of contractible topological space
Topics
About:
topological space
The table of contents of this article
Starting Context
Target Context
-
The reader will have a definition of contractible topological space.
Orientation
There is a list of definitions discussed so far in this site.
There is a list of propositions discussed so far in this site.
Main Body
1: Structured Description
Here is the rules of Structured Description.
Entities:
\(*T\): \(\in \{\text{ the topological spaces }\}\)
//
Conditions:
\(\exists f_c: T \to T, \in \{\text{ the constant maps }\}\)
(
\(id \simeq f_c\), where \(id\) is the identity map, \(id: T \to T\)
).
//
2: Natural Language Description
Any topological space, \(T\), on which the identity map, \(id: T \to T\), is homotopic to a constant map, \(f_c: T \to T\)
References
<The previous article in this series | The table of contents of this series | The next article in this series>
<The previous article in this series | The table of contents of this series | The next article in this series>
definition of homotopic maps
Topics
About:
topological space
The table of contents of this article
Starting Context
Target Context
-
The reader will have a definition of homotopic maps.
Orientation
There is a list of definitions discussed so far in this site.
There is a list of propositions discussed so far in this site.
Main Body
1: Structured Description
Here is the rules of Structured Description.
Entities:
\( T_1\): \(\in \{\text{ the topological spaces }\}\)
\( T_2\): \(\in \{\text{ the topological spaces }\}\)
\(*f\): \(: T_1 \to T_2\), \(\in \{\text{ the continuous maps }\}\)
\(*f'\): \(: T_1 \to T_2\), \(\in \{\text{ the continuous maps }\}\)
//
Conditions:
\(\exists F: T_1 \times I \to T_2, \in \{\text{ the continuous maps }\}\), where \(I\) is \([0, 1] \subseteq \mathbb{R}\)
(
\(\forall p \in T_1\)
(
\(F (p, 0) = f (p)\)
\(\land\)
\(F (p, 1) = f' (p)\)
)
).
//
\(F\) is called "homotopy".
\(f \simeq f'\) denotes the relation.
2: Natural Language Description
For any topological spaces, \(T_1, T_2\), any continuous maps, \(f, f': T_1 \to T_2\), such that there is a continuous map (called "homotopy"), \(F: T_1 \times I \to T_2\), where \(I\) is \([0, 1] \subseteq \mathbb{R}\), such that \(F (p, 0) = f (p)\) and \(F (p, 1) = f' (p)\), denoted as \(f \simeq f'\)
References
<The previous article in this series | The table of contents of this series | The next article in this series>
<The previous article in this series | The table of contents of this series | The next article in this series>
definition of maps homotopic relative to subset of domain
Topics
About:
topological space
The table of contents of this article
Starting Context
Target Context
-
The reader will have a definition of maps homotopic relative to subset of domain.
Orientation
There is a list of definitions discussed so far in this site.
There is a list of propositions discussed so far in this site.
Main Body
1: Structured Description
Here is the rules of Structured Description.
Entities:
\( T_1\): \(\in \{\text{ the topological spaces }\}\)
\( T_2\): \(\in \{\text{ the topological spaces }\}\)
\( S\): \(\subseteq T_1\)
\(*f\): \(: T_1 \to T_2\), \(\in \{\text{ the continuous maps }\}\)
\(*f'\): \(: T_1 \to T_2\), \(\in \{\text{ the continuous maps }\}\)
//
Conditions:
\(\exists F: T_1 \times I \to T_2, \in \{\text{ the continuous maps }\}\), where \(I\) is \([0, 1] \subseteq \mathbb{R}\)
(
\(\forall p \in T_1\)
(
\(F (p, 0) = f (p)\)
\(\land\)
\(F (p, 1) = f' (p)\)
)
\(\land\)
\(\forall p \in S, \forall s \in I\)
(
\(F (p, s) = f (p) = f' (p)\)
)
).
//
\(F\) is called "relative homotopy".
\(f \simeq f' rel S\) denotes the relation.
2: Natural Language Description
For any topological spaces, \(T_1, T_2\), and any subset, \(S \subseteq T_1\), any continuous maps, \(f, f': T_1 \to T_2\), such that there is a continuous map (called "relative homotopy"), \(F: T_1 \times I \to T_2\), where \(I\) is \([0, 1] \subseteq \mathbb{R}\), such that for each \(p \in T_1\), \(F (p, 0) = f (p)\) and \(F (p, 1) = f' (p)\), and for each \(p \in S\) and each \(s \in I\), \(F (p, s) = f (p) = f' (p)\), denoted as \(f \simeq f' rel S\)
References
<The previous article in this series | The table of contents of this series | The next article in this series>
<The previous article in this series | The table of contents of this series | The next article in this series>
description/proof of that covering map into simply connected topological space is homeomorphism
Topics
About:
topological space
The table of contents of this article
Starting Context
Target Context
-
The reader will have a description and a proof of the proposition that any covering map into any simply connected topological space is a homeomorphism.
Orientation
There is a list of definitions discussed so far in this site.
There is a list of propositions discussed so far in this site.
Main Body
1: Structured Description
Here is the rules of Structured Description.
Entities:
\(T_1\): \(\in \{\text{ the connected topological spaces }\} \cap \{\text{ the locally path-connected topological spaces }\}\)
\(T_2\): \(\in \{\text{ the connected topological spaces }\} \cap \{\text{ the locally path-connected topological spaces }\} \cap \{\text{ the simply connected topological spaces }\}\)
\(\pi\): \(:T_1 \to T_2\), \(\in \{\text{ the covering maps }\}\)
//
Statements:
\(\pi \in \{\text{ the homeomorphisms }\}\).
//
2: Natural Language Description
For any connected and locally path-connected topological spaces, \(T_1, T_2\), where \(T_2\) is also simply connected, any covering map, \(\pi: T_1 \to T_2\), which means that \(\pi\) is continuous and surjective and around any point, \(p \in T_2\), there is a neighborhood, \(N_p \subseteq T_2\), that is evenly covered by \(\pi\), is a homeomorphism.
3: Proof
For any point \(p \in T_2\), let us suppose that there were 2 points, \(p_1, p_2 \in \pi^{-1} (p)\), \(p_1 \neq p_2\). \(p_1\) and \(p_2\) would not be path-connected, because if there was a path, \(\lambda: I \to T_1\), such that \(\lambda (0) = p_1\) and \(\lambda (1) = p_2\), \(\pi \circ \lambda\) would be a loop on \(T_2\) and \(\pi \circ \lambda \simeq c_p\) where \(c_p\) would be the constant path into \(\{p\}\), because \(T_2\) would be simply connected; there would be the unique lifts, \(\widetilde{\pi \circ \lambda}\) such that \(\widetilde{\pi \circ \lambda} (0) = p_1\) and \(\widetilde{c_p}\) such that \(\widetilde{c_p} (0) = p_1\), by the proposition that for any covering map, there is the unique lift of any path for each point in the covering map preimage of the path image of any point on the path domain, and \(\widetilde{\pi \circ \lambda} \simeq \widetilde{c_p}\) and \(\widetilde{\pi \circ \lambda} (1) = \widetilde{c_p} (1)\), by the proposition that the lifts, that start at any same point, of any path-homotopic paths are path-homotopic; but \(\widetilde{\pi \circ \lambda} = \lambda\) and \(\widetilde{c_p} = c_{p_1}\), and \(\widetilde{\pi \circ \lambda} (1) = \lambda (1) = p_2\) and \(\widetilde{c_p} (1) = c_{p_1} (1) = p_1\), a contradiction.
So, there would be the path-connected component around each point, \(p_\alpha \in \pi^{-1} (p)\), \(S_\alpha\). There would be no point on \(T_1\) that did not belong to any \(S_\alpha\), because for any point, \(p' \in T_1\), \(p\) and \(\pi (p')\) would be path-connected (because \(T_2\) would be simply connected, which would imply being path-connected), so, there would be a loop, \(l_{\pi (p'), p}: I \to T_2\), such that \(l_{\pi (p'), p} (0) = l_{\pi (p'), p} (1) = \pi (p')\) and \(l_{\pi (p'), p} (1/2) = p\); there would be the unique lift, \(\widetilde{l_{\pi (p'), p}}\), such that \(\widetilde{l_{\pi (p'), p}} (0) = p'\), by the proposition that for any covering map, there is the unique lift of any path for each point in the covering map preimage of the path image of any point on the path domain, and \(\widetilde{l_{\pi (p'), p}}\) would pass one of \(\pi^{-1} (p)\), so, \(p'\) would be path-connected with one of \(\pi^{-1} (p)\).
So, \(T_1 = \cup_\alpha S_\alpha\), but each \(S_\alpha\) would be open, by the proposition that any path-connected topological component is open and closed on any locally path-connected topological space, and \(\{S_\alpha\}\) would be disjoint, so, if there were multiple points in \(\pi^{-1} (p)\), \(T_1\) would not be connected. So, there is only 1 point in \(\pi^{-1} (p)\). So, \(\pi\) is a bijection.
So, \(\pi^{-1}\) is a map, which is continuous, because around each point, \(p \in T_2\), there is an open neighborhood, \(U_p \subseteq T_2\), such that \(\pi^{-1} \vert_{U_p}: U_p \to \pi^{-1} (U_p)\) is continuous, by the proposition that any map between topological spaces is continuous if the domain restriction of the map to each open set of a possibly uncountable open cover is continuous.
References
<The previous article in this series | The table of contents of this series | The next article in this series>
<The previous article in this series | The table of contents of this series | The next article in this series>
definition of lift of continuous map by covering map
Topics
About:
topological space
The table of contents of this article
Starting Context
Target Context
-
The reader will have a definition of lift of continuous map by covering map.
Orientation
There is a list of definitions discussed so far in this site.
There is a list of propositions discussed so far in this site.
Main Body
1: Structured Description
Here is the rules of Structured Description.
Entities:
\( T_1\): \(\in \{\text{ the connected topological spaces }\} \cap \{\text{ the locally path-connected topological spaces }\}\)
\( T_2\): \(\in \{\text{ the connected topological spaces }\} \cap \{\text{ the locally path-connected topological spaces }\}\)
\( \pi\): \(: T_1 \to T_2\), \(\in \{\text{ the covering maps }\}\)
\( T_3\): \(\in \{\text{ the topological spaces }\}\)
\( f\): \(: T_3 \to T_2\), \(\in \{\text{ the continuous maps }\}\)
\(*\tilde{f}\): \(: T_3 \to T_1\), \(\in \{\text{ the continuous maps }\}\)
//
Conditions:
\(f = \pi \circ \tilde{f}\).
//
2: Natural Language Description
For any connected and locally path-connected topological spaces, \(T_1, T_2\), any covering map, \(\pi: T_1 \to T_2\), any topological space, \(T_3\), and any continuous map, \(f: T_3 \to T_2\), any continuous map, \(\tilde{f}: T_3 \to T_1\), such that \(f = \pi \circ \tilde{f}\)
References
<The previous article in this series | The table of contents of this series | The next article in this series>
<The previous article in this series | The table of contents of this series | The next article in this series>
definition of simply connected topological space
Topics
About:
topological space
The table of contents of this article
Starting Context
Target Context
-
The reader will have a definition of simply connected topological space.
Orientation
There is a list of definitions discussed so far in this site.
There is a list of propositions discussed so far in this site.
Main Body
1: Structured Description
Here is the rules of Structured Description.
Entities:
\(*T\): \(\in \{\text{ the path-connected topological spaces }\}\)
//
Conditions:
\(\forall p \in T (\pi_1 (T, p) = \{1\})\) where \(\pi_1 (T, p)\) is the fundamental group.
//
2: Natural Language Description
Any path-connected topological space, \(T\), such that at each point, \(p \in T\), the fundamental group is \(\pi_1 (T, p) = \{1\}\)
References
<The previous article in this series | The table of contents of this series | The next article in this series>
<The previous article in this series | The table of contents of this series | The next article in this series>
definition of linear map
Topics
About:
vectors space
The table of contents of this article
Starting Context
Target Context
-
The reader will have a definition of linear map.
Orientation
There is a list of definitions discussed so far in this site.
There is a list of propositions discussed so far in this site.
Main Body
1: Structured Description
Here is the rules of Structured Description.
Entities:
\( F\): \(\in \{\text{ the fields }\}\)
\( V_1\): \(\in \{\text{ the vectors spaces over } F\}\)
\( V_2\): \(\in \{\text{ the vectors spaces over } F\}\)
\(*f\): \(: V_1 \to V_2\)
//
Conditions:
\(\forall r_1, r_2 \in F, \forall v_1, v_2 \in V_1 (f (r_1 v_1 + r_2 v_2) = r_1 f (v_1) + r_2 f (v_2))\).
//
2: Natural Language Description
For any field, \(F\), and any vectors spaces, \(V_1, V_2\), over \(F\), any map, \(f: V_1 \to V_2\), such that for any \(r_1, r_2 \in F\) and any \(v_1, v_2 \in V_1\), \(f (r_1 v_1 + r_2 v_2) = r_1 f (v_1) + r_2 f (v_2)\)
3: Note
The fields of \(V_1\) and \(V_2\) have to be the same in order for \(r_1 f (v_1) + r_2 f (v_2)\) to make sense; there may be an argument that the field of \(V_2\) could be a superset of that of \(V_1\), but we do not see any immediate necessity to include that case. For example, the complex numbers field is not exactly any superset of the real numbers field (\(1 + 0 i \in \mathbb{C}\) and \(1 \in \mathbb{R}\) are not exactly same according to the standard definitions: intuitively speaking, \((1, 0)\) and \(1\) are not same), although there is the canonical map from \(\mathbb{R}\) into \(\mathbb{C}\).
The \(0\) vector is inevitably mapped to the \(0\) vector, because \(f (0) = f (0 v) = 0 f (v) = 0\) for any \(v \in V_1\).
Any linear map is nothing but a 'vectors spaces' homomorphism.
References
<The previous article in this series | The table of contents of this series | The next article in this series>
<The previous article in this series | The table of contents of this series | The next article in this series>
definition of topological subspace
Topics
About:
topological space
The table of contents of this article
Starting Context
Target Context
-
The reader will have a definition of topological subspace.
Orientation
There is a list of definitions discussed so far in this site.
There is a list of propositions discussed so far in this site.
Main Body
1: Structured Description
Here is the rules of Structured Description.
Entities:
\( T'\): \(\in \{\text{ the topological spaces }\}\)
\(*T\): \(\subseteq T'\) with the subspace topology
//
Conditions:
//
2: Natural Language Description
For any topological space, \(T'\), any subset, \(T \subseteq T'\), with the subspace topology
References
<The previous article in this series | The table of contents of this series | The next article in this series>
<The previous article in this series | The table of contents of this series | The next article in this series>
definition of Euclidean metric
Topics
About:
metric space
The table of contents of this article
Starting Context
Target Context
-
The reader will have a definition of Euclidean metric.
Orientation
There is a list of definitions discussed so far in this site.
There is a list of propositions discussed so far in this site.
Main Body
1: Structured Description
Here is the rules of Structured Description.
Entities:
\( \mathbb{R}^d\): \(= \text{ the Euclidean vectors space with the Euclidean norm }\)
\(*dist\): \(= \text{ the metric induced by the Euclidean norm }\)
//
Conditions:
//
2: Natural Language Description
For the Euclidean set, \(\mathbb{R}^d\), the metric induced by the Euclidean norm on the Euclidean vectors space
3: Note
The Euclidean metric space, \(\mathbb{R}^d\), does not need to really have the Euclidean vectors space structure or the Euclidean norm: the norm is used just in order to define the metric, and the norm and the vectors space structure can be forgotten after that if one likes so. In fact, the metric can be defined without the norm, but this definition uses the norm in order to use the fact that the metric induced by any norm is indeed a metric.
References
<The previous article in this series | The table of contents of this series | The next article in this series>
<The previous article in this series | The table of contents of this series | The next article in this series>
definition of topology induced by metric
Topics
About:
topological space
The table of contents of this article
Starting Context
Target Context
-
The reader will have a definition of topology induced by metric.
Orientation
There is a list of definitions discussed so far in this site.
There is a list of propositions discussed so far in this site.
Main Body
1: Structured Description
Here is the rules of Structured Description.
Entities:
\( T\): \(\in \{\text{ the metric spaces }\}\)
\(*O\): \(\in \{\text{ the topologies of } T\}\)
//
Conditions:
\(\forall S \subseteq T (S \in O \iff (\forall p \in S (\exists \epsilon \in \mathbb{R} (0 \lt \epsilon \land (B_{p, \epsilon} \subseteq S)))))\).
//
2: Natural Language Description
For any metric space, \(T\), the topology, \(O\), such that for any subset, \(S \subseteq T\), \(S\) is open if and only if for each \(p \in S\), there is a positive, \(\epsilon \in \mathbb{R}\), such that \(B_{p, \epsilon} \subseteq S\)
3: Note
It is indeed a topology: \(\emptyset \in O\); \(T \in O\); for any possibly uncountable number of open subsets, \(\{U_\alpha \vert \alpha \in A\}\), and \(U := \cup_{\alpha \in A} U_\alpha\), for any \(p \in U\), \(B_{p, \epsilon_\alpha} \subseteq U_\alpha\) for any fixed \(\alpha\), and \(B_{p, \epsilon_\alpha} \subseteq U_\alpha \subseteq U\); for any finite number of open subsets, \(U_1, ..., U_k\), and \(U := \cap_{j = 1, .., k} U_k\), for any \(p \in U\), \(B_{p, \epsilon_j} \subseteq U_j\) for each \(j\), and for \(\epsilon := \min \{\epsilon_j\}\), \(0 \lt \epsilon\) and \(B_{p, \epsilon} \subseteq U\).
For each \(p \in T\) and each \(\epsilon\), \(B_{p, \epsilon} \in O\), because for any \(p' \in B_{p, \epsilon}\), \(dist (p, p') \le \epsilon\), so, for \(B_{p', \epsilon - dist (p, p')}\), for each \(p'' \in B_{p', \epsilon - dist (p, p')}\), \(dist (p'', p') \lt \epsilon - dist (p, p')\), but \(dist (p'', p) \le dist (p'', p') + dist (p', p) \lt \epsilon - dist (p, p') + dist (p', p) = \epsilon\), which means that \(B_{p', \epsilon - dist (p, p')} \subseteq B_{p, \epsilon}\).
References
<The previous article in this series | The table of contents of this series | The next article in this series>
<The previous article in this series | The table of contents of this series | The next article in this series>
description/proof of that square of Euclidean norm of \(\mathbb{R}^n\) vector is equal to or smaller than positive definite real quadratic form divided by smallest eigenvalue
Topics
About:
vectors space
The table of contents of this article
Starting Context
Target Context
-
The reader will have a description and a proof of the proposition that for any \(\mathbb{R}^n\) vectors space, the square of the Euclidean norm of any vector is equal to or smaller than any positive definite real quadratic form divided by the smallest eigenvalue of the quadratic form.
Orientation
There is a list of definitions discussed so far in this site.
There is a list of propositions discussed so far in this site.
Main Body
1: Structured Description
Here is the rules of Structured Description.
Entities:
\(\mathbb{R}^d\): with the Euclidean vectors space structure and the Euclidean norm, \(\Vert v \Vert = \sqrt{{v^1}^2 + {v^2}^2 + ... + {v^d}^2}\)
\(M\): \(\in \{\text{ the real symmetric } d \times d \text{ matrices }\}\)
\(f\): \(\in \{\text{ the positive definite real quadratic forms over } \mathbb{R}^d\}\), \(: \mathbb{R}^d \to \mathbb{R}\), \(v \mapsto v^t M v\)
\(v\): \(\in \mathbb{R}^d\)
\(\lambda_m\): \(= \text{ the smallest eigenvalue of } M\), inevitably \(0 \lt \lambda_m\)
//
Statements:
\(\Vert v \Vert^2 \leq f (v) / \lambda_m\).
//
2: Natural Language Description
For the \(\mathbb{R}^n\) Euclidean vectors space with the Euclidean norm, \(\Vert v \Vert = \sqrt{{v^1}^2 + {v^2}^2 + ... + {v^d}^2}\), and any positive definite real quadratic form, \(f (v) = v^t M v\), where \(M\) is a real symmetric \(d \times d\) matrix, \(\Vert v \Vert^2 \leq f (v) / \lambda_m\), where \(\lambda_m\) (inevitably \(0 \lt \lambda_m\)) is the smallest eigenvalue of \(M\).
3: Proof
There is an orthonormal matrix, \(M'\), such that \(M'^{-1} M M' = [\lambda_1, ..., \lambda_d]\) where \([\lambda_1, ..., \lambda_d]\) is a diagonal matrix where \(\lambda_i\) is an eigenvalue, by the proposition that any real symmetric matrix can be diagonalized by an orthonormal matrix. Let us define \(v' := M'^{-1} v\), so, \(v = M' v'\). \(f (v) = v'^{t} M'^{t} M M' v' = v'^{t} M'^{-1} M M' v' = v'^{t} [\lambda_1, ..., \lambda_d] v' = \lambda_1 {v'^1}^2 + \lambda_2 {v'^2}^2 + ... + \lambda_d {v'^d}^2\), by the proposition that the transposition of any orthonormal matrix is the inverse of the original matrix
Let us denote the smallest eigenvalue as \(\lambda_m\), which is inevitably \(0 \lt \lambda_m\), because \(M\) is positive definite. \(\lambda_m \Vert v' \Vert^2 = \lambda_m ({v'^1}^2 + {v'^2}^2 + ... + {v'^d}^2) \leq \lambda_1 {v'^1}^2 + \lambda_2 {v'^2}^2 + ... + \lambda_d {v'^d}^2 = f (v)\). As \(M'\) is an orthonormal matrix, \(\Vert v' \Vert = \Vert v \Vert\). So, \(\Vert v \Vert^2 \leq f (v) / \lambda_m\).
References
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