Highest Reality. Thought of the Day 70.0


यावचिन्त्यावात्मास्य शक्तिश्चैतौ परमार्थो भवतः॥१॥

Yāvacintyāvātmāsya śaktiścaitau paramārtho bhavataḥ

These two (etau), the Self (ātmā) and (ca) His (asya) Power (śaktiḥ) —who (yau) (are) inconceivable (acintyau)—, constitute (bhavataḥ) the Highest Reality (parama-arthaḥ)

The Self is the Core of all, and His Power has become all. I call the Core “the Self” for the sake of bringing more light instead of more darkness. If I had called Him “Śiva”, some people might consider Him as the well-known puranic Śiva who is a great ascetic living in a cave and whose main task consists in destroying the universe, etc. Other people would think that, as Viṣṇu is greater than Śiva, he should be the Core of all and not Śiva. In turn, there is also a tendency to regard Śiva like impersonal while Viṣṇu is personal. There is no end to spiritual foolishness indeed, because there is no difference between Śiva and Viṣṇu really. Anyway, other people could suggest that a better name would be Brahman, etc. In order not to fall into all that ignorant mess of names and viewpoints, I chose to assign the name “Self” to the Core of all. In the end, when spiritual enlightenment arrives, one’s own mind is withdrawn (as I will tell by an aphorism later on), and consequently there is nobody to think about if “This Core of all” is personal, impersonal, Śiva, Viṣṇu, Brahman, etc. Ego just collapses and This that remains is the Self as He essentially is.

He and His Power are completely inconceivable, i.e. beyond the mental sphere. The Play of names, viewpoints and such is performed by His Power, which is always so frisky. All in all, the constant question is always: “Is oneself completely free like the Self?”. If the answer is “Yes”, one has accomplished the goal of life. And if the answer is “No”, one must get rid of his own bondage somehow then. The Self and His Power constitute the Highest Reality. Once you can attain them, so to speak, you are completely free like Them both. The Self and His Power are “two” only in the sphere of words, because as a matter of fact they form one compact mass of Absolute Freedom and Bliss. Just as the sun can be divided into “core of the sun, surface of the sun, crown”, etc.

तयोरुभयोः स्वरूपं स्वातन्त्र्यानन्दात्मकैकघनत्वेनापि तत्सन्तताध्ययनाय वचोविषय एव द्विधाकृतम्

Tayorubhayoḥ svarūpaṁ svātantryānandātmakaikaghanatvenāpi tatsantatādhyayanāya vacoviṣaya eva dvidhākṛtam

Even though (api) the essential nature (sva-rūpam) of Them (tayoḥ) both (ubhayoḥ) (is) one compact mass (eka-ghanatvena) composed of (ātmaka) Absolute Freedom (svātantrya)(and) Bliss (ānanda), it is divided into two (dvidhā-kṛtam) —only (eva) in the sphere (viṣaye) of words (vacas)— for its close study (tad-santata-adhyayanāya)

The Self is Absolute Freedom and His Power is Bliss. Both form a compact mass (i.e. omnipresent). In other words, the Highest Reality is always “One without a second”, but, in the world of words It is divided into two for studying It in detail. When this division occurs, the act of coming to recognize or realize the Highest Reality is made easier. So, the very Highest Reality generates this division in the sphere of words as a help for the spiritual aspirants to realize It faster.

आत्मा प्रकाशात्मकशुद्धबोधोऽपि सोऽहमिति वचोविषये स्मृतः

Ātmā prakāśātmakaśuddhabodho’pi so’hamiti vacoviṣaye smṛtaḥ

Although (api) the Self (ātmā) (is) pure (śuddha) Consciousness (bodhaḥ) consisting of (ātmaka) Prakāśa or Light (prakāśa), He (saḥ) is called (smṛtaḥ) “I” (aham iti) in the sphere (viṣaye) of words (vacas)

The Self is pure Consciousness, viz. the Supreme Subject who is never an object. Therefore, He cannot be perceived in the form of “this” or “that”. He cannot even be delineated in thought by any means. Anyway, in the world of words, He is called “I” or also “real I” for the sake of showing that He is higher than the false “I” or ego.

Banach Spaces


Some things in linear algebra are easier to see in infinite dimensions, i.e. in Banach spaces. Distinctions that seem pedantic in finite dimensions clearly matter in infinite dimensions.

The category of Banach spaces considers linear spaces and continuous linear transformations between them. In a finite dimensional Euclidean space, all linear transformations are continuous, but in infinite dimensions a linear transformation is not necessarily continuous.

The dual of a Banach space V is the space of continuous linear functions on V. Now we can see examples of where not only is V* not naturally isomorphic to V, it’s not isomorphic at all.

For any real p > 1, let q be the number such that 1/p  + 1/q = 1. The Banach space Lp is defined to be the set of (equivalence classes of) Lebesgue integrable functions f such that the integral of ||f||p is finite. The dual space of Lp is Lq. If p does not equal 2, then these two spaces are different. (If p does equal 2, then so does qL2 is a Hilbert space and its dual is indeed the same space.)

In the finite dimensional case, a vector space V is isomorphic to its second dual V**. In general, V can be embedded into V**, but V** might be a larger space. The embedding of V in V** is natural, both in the intuitive sense and in the formal sense of natural transformations. We can turn an element of V into a linear functional on linear functions on V as follows.

Let v be an element of V and let f be an element of V*. The action of v on f is simply fv. That is, v acts on linear functions by letting them act on it.

This shows that some elements of V** come from evaluation at elements of V, but there could be more. Returning to the example of Lebesgue spaces above, the dual of L1 is L, the space of essentially bounded functions. But the dual of L is larger than L1. That is, one way to construct a continuous linear functional on bounded functions is to multiply them by an absolutely integrable function and integrate. But there are other ways to construct linear functionals on L.

A Banach space V is reflexive if the natural embedding of V in V** is an isomorphism. For p > 1, the spaces Lp are reflexive.

Suppose that X is a Banach space. For simplicity, we assume that X is a real Banach space, though the results can be adapted to the complex case in the straightforward manner. In the following, B(x0,ε) stands for the closed ball of radius ε centered at x0 while B◦(x0,ε) stands for the open ball, and S(x0,ε) stands for the corresponding sphere.

Let Q be a bounded operator on X. Since we will be interested in the hyperinvariant subspaces of Q, we can assume without loss of generality that Q is one-to-one and has dense range, as otherwise ker Q or Range Q would be hyperinvariant for Q. By {Q}′ we denote the commutant of Q.

Fix a point x0 ≠ 0 in X and a positive real ε<∥x0∥. Let K= Q−1B(x0,ε). Clearly, K is a convex closed set. Note that 0 ∉ K and K≠ ∅ because Q has dense range. Let d = infK||z||, then d > 0. If X is reflexive, then there exists z ∈ K with ||z|| = d, such a vector is called a minimal vector for x0, ε and Q. Even without reflexivity condition, however, one can always find y ∈ K with ||y|| ≤ 2d, such a y will be referred to as a 2-minimal vector for x0, ε and Q.

The set K ∩ B(0, d) is the set of all minimal vectors, in general this set may be empty. If z is a minimal vector, since z ∈ K = Q−1B(x0, ε) then Qz ∈ B(x0, ε). As z is an element of minimal norm in K then, in fact, Qz ∈ S(x0, ε). Since Q is one-to-one, we have

QB(0, d) ∩ B(x0, ε) = Q B(0, d) ∩ K) ⊆ S(x0, ε).

It follows that QB(0,d) and B◦(x0,ε) are two disjoint convex sets. Since one of them has non-empty interior, they can be separated by a continuous linear functional. That is, there exists a functional f with ||f|| = 1 and a positive real c such that f|QB(0,d)  ≤ c and f|B◦(x0,ε) ≥ c. By continuity, f|B(x0,ε) ≥ c. We say that f is a minimal functional for x0, ε, and Q.

We claim that f(x0) ≥ ε. Indeed, for every x with ||x|| ≤ 1 we have x0 − εx ∈ B(x0,ε). It follows that f(x0 − εx) ≥ c, so that f(x0) ≥ c + εf(x). Taking sup over all x with ||x|| ≤ 1 we get f(x0) ≥ c + ε||f|| ≥ ε.

Observe that the hyperplane Qf = c separates K and B(0, d). Indeed, if z ∈ B(0,d), then (Qf)(z) = f(Qz) ≤ c, and if z ∈ K then Qz ∈ B(x0,ε) so that (Q∗f)(z) = f(Qz) ≥ c. For every z with ||z|| ≤ 1

we have dz ∈ B(0,d), so that (Qf)(dz) ≤ c, it follows that Qf ≤ c/d

On the other hand, for every δ > 0 there exists z ∈ K with ||z|| ≤ d+δ, then (Qf)(z) ≥ c ≥ c/(d+δ) ||z||, whence ||Qf|| ≥ c/(d+δ) . It follows that

||Q∗f|| = c/d.

For every z ∈ K we have (Qf)(z) ≥ c = d ||Qf||. In particular, if y is a 2-minimal vector then

(Qf)(y) ≥ 1/2 Qf ||y||….