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**Problems based on Module â€“I (Metric Spaces) **

**Ex.1.**** Let d be a metric on X. Determine all constants K such that(i) kd , (ii) d + k is a metric on XHint.1:** Use definition . Ans (i) k>0 (ii) k=0

**Ex.2. Show thatHint.2: **Use Triangle inequality d

Similarity,

**Ex.3. Find a sequence x which is in l ^{p} with p >1 but Hint.3:** Choose x = (x

**Ex.4. Let (X,d) be a metric space and A,B are any two non empty subsets of X. Is D(A,B) = inf d (a,b)a metric on the power set of X? Hint.4:** No. Because D(A,B)=0 â‰ > A=B e.g. Choose where a

**Ex.5. Let ( X, d) be any metric space. Is a Metric space where Hint.5:** Yes. ;

**Ex.6. Let (X _{1}, d_{1}) and (X_{2}, d_{2}) be metric Spaces and X = X_{1} X_{2}. Are as defined below A metric on X?** Yes.

Hint.6:

**Ex.7. Show that in a discrete metric space X, every subset is open and closed.Hint.7:** Any subset is open since for any a âˆˆ A, the open ball Similarly A

**Ex.8. Describe the closure of each of the following Subsets:(a) The integers on R.(b) The rational numbers on R.(c) The complex number with real and imaginary parts as rational in.(d) The disk Hint.8:** use Definition

Ans (a) The integer, (b) R, (c)

**Ex.9. Show that a metric space X is separable if and only if X has a countable subset Y with the property: For every âˆˆ > 0 and every x âˆˆ X there is a y âˆˆ Y such that d (x ,y) < âˆˆ.Hint.9: **Let X be separable .So it has a countable dense subset Y i.e. be given.Since Y is dense in X and so that the âˆˆ neibourhood B(x;âˆˆ) of x contains a y âˆˆ Y, and d (x, y) < âˆˆ. Conversely, if X has a countable subset Y with the property given in the problem, every x âˆˆ X is a point of Y or an accumulation point of Y. Hence x âˆˆ Y, s result follows.

**Ex.10. If (x _{n}) and (y_{n}) are Cauchy sequences in a metric space (X,d), show that (a_{n}), where a_{n }= d(x_{n}.y_{n}) converges.** Since

Hint.10:

which shows that (a

**Ex.11. Let ab âˆˆ R and a < b.Show that the open interval (a, b) is an incomplete subspace of R.Hint.11: **Choose which is a Cauchy sequence in (a,b) but does not converge.

**Ex.12. Let be the subspace consisting of all sequence with at most finitely many nonzero terms .Find a Cauchy sequence in M which does not converge in M, so that M is not complete.Hint.12: **Choose (x

**Ex.13. Show that the set X of all integers with metric d defined by d (m,n) = |m - n| is a complete metric space.Hint.**

Where x

**Ex.14.Let X be the set of all positive integers and d (m, n) = |m ^{-1} - n^{-1}|. Show that (X , d) is not complete.** Choose (x

Hint.14:

**Ex.15. Show that a discrete metric space is complete.Hint.15: **Constant sequence are Cauchy and convergent.

**Ex.16. Let X be metric space of all real sequences each of which has only finitely Nonzero terms, and when y = (Î· _{j}). Show that **

Hint.16:

But (x

And d(x

**Ex.17. Show that,by given a example ,that a complete and an incomplete metric spaces may be Homeomorphic.Hint.17:** (Def) A homeomorphism is a continuous bijective mapping.

T: X â†’ Y : whose inverse is continuous; the metric space X and Y are then said to be homeomorphic. e g. . A mapping T: R: â†’ (-1 ,1) defined as with metric d(x,y) = |x - y| on R. .Clearly T is 1-1,into & bi continuous so But R is complete while (-1,1) is an incomplete metric space.

**Ex.18. If (X , d) is complete, show that where is complete.Hint.18:**

Hence if (x

**Ex.1. Let (X, i= 1, 2, âˆž be normed spaces of all ordered pairs of real numbers where are defined as **

**Ex.2. Show that the discrete metric on a vector space X â‰ {0} can not be obtained ****from a norm.****Hint.2:**** **

**Ex.3. In l ^{âˆž}**

as well as x^{(n)}âˆˆ Î³ but

**Ex.4. Give examples of subspaces of l ^{âˆž} and l^{2} which are not closed.** Let Î³ be the subset of all sequences with only finitely many non zero terms.

Hint.4:

but not closed.

**Ex.5. Show that a discrete metric space X consisting of infinitely many points is not ****compact.Hint.5:** By def. of Discrete metric, any sequences (n

**Ex.6. Give examples of compact and non compact curves in the plane R ^{2}.Hint.6: **As R

Choose which is compact But is not compact.

**Ex.7. Show that are locally compact. Hint.7:** (def.) A metric space X is said to be locally compact if every point of X has a compact neighbourhood. Result follows (obviously).

**Ex.8. Let X and Y be metric spaces. X is compact and T: Xâ†’ Y bijective and ****continuous. Show that T is homeomorphism.Hint.8:** Only to show T

OR.

If It will follow from the fact that X is compact.

**Ex.9. What are the domain, range and null space of T _{1},T_{2},T_{3} in exercise 9.** The domain is R

Hint.9:

**Ex.10. Let T : X â†’ Y be a linear operator. Show that the image of a subspace V of X ****is a vector space, and so is the inverse image of a subspace W of X.Hint.10.**

Î±x

**Ex.11. Let X be the vector space of all complex 2 x 2 matrices and define T: X â†’ X by Tx=bx, where b âˆˆ X is fixed and bx denotes the usual product of matrices. Show that T is linear. Under what condition does T ^{-1} exist?Hint.11.** |b| â‰ 0

**Ex.12. Let T : D(T) â†’ Y be a linear operator whose inverse exists. If {x _{1},x_{2},....,x_{n}) **

Hint.12.

**Ex.13. Consider the vector space X of all real-valued functions which are defined on ****R and have derivatives of all orders everywhere on R. Define T : X â†’ X by y(t) = Tx(t) = x'(t), show that R(T) is all of X but T ^{-1 } does not exist.Hint.13:** R (T) = X since for every y âˆˆ X we have y = Tx, where x(t) = But n T

**Ex.14. Let X and Y be normed spaces. Show that a linear operator T: X â†’ Y is bounded if and only if T maps bounded sets in X into bounded set in Y.Hint.14:** Apply definition of bounded operator.

**Ex.15. If T â‰ 0 is a bounded linear operator, show that for any we have the strict inequality Hint.15:** Since

**Ex.16. Find the norm of the linear functional f defined on C[-1, 1] byHint.16: **For converse, choose x(t) = -1 on [-1,1]. So

**Problems on Module III (IPS/Hilbert space)**

**Ex.1. If x âŠ¥ y in an IPS X,Show that ****Hint.1: **Use and the fact that < x y > = 0, if x âŠ¥ y.

**Ex.2. If X in exercise 1 is a real vector space, show that ,conversely, the given relation implies that x âŠ¥ y . Show that this may not hold if X is complex. Give examples.Hint. 2:** By Assumption,

**Ex.3. If an IPS X is real vector space, show that the condition implies <x +y,x-y>= 0. What does this mean geometrically if X = R ^{2}?Hint.3:** Start <x +y,x-y> = <x,x> + <y,-y> = as X is real. Geometrically: If x & y are the vectors representing the sides of a parallelogram, then x+y and x-y will represent the diagonal which areâŠ¥.

**Ex.4. (Apollonius identity): For any elements x, y, z in an IPS X, show thatHint 4:** Use OR use parallelogram equality.

**Ex.5. Let x â‰ 0 and y â‰ 0. If x âŠ¥ y, show that {x,y} is a Linearly Independent set.Hint.5:** Suppose where Î±

.Similarly, one can show that is L.I.set.

**Ex.6. If in an IPS X, <x,u> = <x,v> for all x, show that u = v.Hint.6 :** Given <x,u-v> = 0. Choose x = u - v.

**Ex.7. Let X be the vector space of all ordered pairs of complex numbers. Can we obtain **** the norm defined on X by from an Inner product?Hint. 7:** No. because the vectors x= (1,1), y = (1,1) do not satisfy parallelogram equality.

**Ex.8. If X is a finite dimensional vector space and (e _{j} ) is a basis for X, show that an inner product on X is completely determined by its values . Can we choose scalars Î³_{jk} in a completely arbitrary fashion?** Use

Hint.8:

Open it so we get that it depends on

II Part: Answer:- NO. Because

**Ex.9. Show that for a sequence (x _{n} ) in an IPS X , the conditionsHint.9 : **We have

**Ex.10. Show that in an IPS X, for all scalars Î±.Hint.10 :** From

condition follows as x âŠ¥ y.

Conversely,

Choose Î± = 1 if the space is real which implies x âŠ¥ y.

Choose Î± = 1, Î± = i if the space is complex then we get <x,y> = 0 â‡’x âŠ¥ y.

**Ex.11. Show that in an IPS X, for all scalars.Hint.11 :** Follows from the hint given in Ex.-10.

**Ex.12. Let V be the vector space of all continuous complex valued functions on J = [a,b]. ****where Show that the identity mapping of X _{1 }onto X_{2 } is continuous. Is it Homeomorphism?**Since

Hint.12 :

Hence I is continuous.

Part-II: Answer No. because X

**Ex.13. Let H be a Hilbert space, a convex subset, and (x _{n}) is a sequence in M such that , where Show that (x_{n}) converges in H.** (x

Hint.13 :

Hint.14 :

**Ex.15. Let (e _{k}) be any orthonormal sequence in an IPS X. Show that for any , x y âˆˆ X, **

**Ex.-16. Show that in a Hilbert Space H,convergence of implies convergence of****Hint.16 : **Let

is a Cauchy. Since H is complete, hence will converge.

converge in H.

**Problems On Module IV (On Fundamental theorems)**

**Ex.1. Let be a sequence of bounded linear functionals defined as where . show that (f _{n})converge strongly to 0 but not uniformly.**

ie

Hint.2

T is bounded :- Since

So (T

**Ex.3. If . Show that (x _{n}) is point wise convergent on [a,b].Hint .3 :** A bounded linear functional on

**Ex.4. If in a normed space X. Show thatHint.4 :** Use Lemma:- â€˜â€™Let Y be a proper closed sub-space of a normed space X and let be arbitrary point and then there exists an , dual of X such that for all y âˆˆ Y and

suppose which is a closed sub space of X. so by the above result ,

hence there exists

which is a contradiction that

**Ex.5. Let T _{n} = S^{n}, where the operator is defined by S Find a bound for **

Hint.5 :

(iv)

**Ex.6. Let X be a Banach space, Y a normed space and such that (T _{n}x) is Cauchy in Y for every x âˆˆ X. .show that is bounded.Hint. 6 : **Since (T

**Ex.7. If (x _{n}) in a Banach space X is such that is bounded for all.Show that is bounded.Hint.7 :** Suppose Thenis bounded for every . So by uniform bounded ness theorem is bounded and .

**Ex.8. If a normed space X is reflexive, Show that X' is reflexive.Hint. 8 :** Let there is an x âˆˆ X such that g = Cx since X is reflexive. Hence h(g) = h (Cx) = f(x) defines a bounded linear functional f on X and where is the canonical mapping. Hence C

**Ex.9. If x _{0} in a normed space X is such that of norm1. show that Hint. 9 :** suppose Then by Lemma: Let X be a normed space and letbe any elementof X. Then there exist a bounded linear functional

would imply the existence of an

**Ex.10. Let Y be a closed sub space of a normed space X such that every f âˆˆ X which is zero every where on Y is zero every where on the whole space X. Show that Y = XHint. 10 : **If , Y â‰ X there is an since Y is closed.

Use the Lemma as given in Ex 4 ( Hint ).

By this Lemma, there is on which is zero on Y but not zero at x

**Ex.11. Prove that Where T ^{x }is the adjoint operator of T.Hint. 11 : **

Similarly others.

**Ex.12. Prove (ST) ^{x }= T^{x}S^{x}**

**Ex.13. Show that (T ^{n})^{x} = (T^{x})^{n}.**

**Ex.14. Of what category is the set of all rational number (a) in , ( b ) in itself, (Taken usual metric)Hint 14 :** (a) first (b) first.

**Ex.15. Find all rare sets in a discrete metric space X. ****Hint.15 : ** because every subset of X is open

**Ex.16. Show that a subset M of a metric space X is rare in X if and only if is is dense in X.Hint. 16 :** The closure of is all of X if and if has no interior points, So that every is a point of accumulation of

**Ex.17. Show that completeness of X is essential in uniform bounded ness theorem and cannot be omitted.Hint.17 : **Consider the sub space consisting of all where J depends on x, and let T

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