Normed Spaces, Banach Spaces and Operators - Functional Analysis - Mathematics Previous Year Questions

1

2008 · Mathematics · Functional Analysis · Normed Spaces, Banach Spaces and Operators

Mathematics (MA) 2008

For \(1 \leq p \leq \infty\), let \(\| \|_p\) denote the \(p\)-norm on \(\mathbb{R}^2\). If \(\| \|_p\) satisfies the parallelogram law, then \(p\) is equal to

A

1

B

2

C

3

D

\(\infty\)

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2

2008 · Mathematics · Functional Analysis · Normed Spaces, Banach Spaces and Operators

Mathematics (MA) 2008

Let \(f: \ell^2 \to \mathbb{R}\) be defined by \(f(x_1, x_2, \cdots) = \sum_{n=1}^{\infty} \frac{x_n}{2^{n/2}}\) for \((x_1, x_2, \cdots) \in \ell^2\). Then \(\|f\|\) is equal to

A

\(\frac{1}{2}\)

B

1

C

2

D

\(\frac{1}{\sqrt{2}-1}\)

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3

2008 · Mathematics · Functional Analysis · Normed Spaces, Banach Spaces and Operators

Mathematics (MA) 2008

Consider \(\mathbb{R}^3\) with norm \(\|\cdot\|_\infty\) and the linear transformation \(T: \mathbb{R}^3 \to \mathbb{R}^3\) defined by the 3x3 matrix \(\begin{pmatrix} 1 & 1 & 3 \\ 2 & 2 & 2 \\ 1 & 3 & -3 \end{pmatrix}\). Then the operator norm \(\|T\|\) of \(T\) is equal to

A

6

B

7

C

8

D

\(\sqrt{42}\)

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4

2008 · Mathematics · Functional Analysis · Normed Spaces, Banach Spaces and Operators

Mathematics (MA) 2008

Consider \(\mathbb{R}^2\) with norm \(\|\cdot\|_\infty\) and let \(Y = \{(y_1, y_2) \in \mathbb{R}^2 : y_1 + y_2 = 0\}\). If \(g: Y \to \mathbb{R}\) is defined by \(g(y_1, y_2) = y_2\) for \((y_1, y_2) \in Y\), then

A

\(g\) has no Hahn-Banach extension to \(\mathbb{R}^2\)

B

\(g\) has a unique Hahn-Banach extension to \(\mathbb{R}^2\)

C

every linear functional \(f: \mathbb{R}^2 \to \mathbb{R}\) satisfying \(f(-1, 1) = 1\) is a Hahn-Banach extension of \(g\) to \(\mathbb{R}^2\)

D

the functionals \(f_1, f_2: \mathbb{R}^2 \to \mathbb{R}\) given by \(f_1(x_1, x_2) = x_2\) and \(f_2(x_1, x_2) = -x_1\) are both Hahn-Banach extensions of \(g\) to \(\mathbb{R}^2\)

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5

2008 · Mathematics · Functional Analysis · Normed Spaces, Banach Spaces and Operators

Mathematics (MA) 2008

Let \(X\) be a Banach space and \(Y\) be a normed linear space. Consider a sequence \((F_n)\) of bounded linear maps from \(X\) to \(Y\) such that for each fixed \(x \in X\), the sequence \((F_n(x))\) is bounded in \(Y\). Then

A

for each fixed \(x \in X\), the sequence \((F_n(x))\) is convergent in \(Y\)

B

for each fixed \(n \in \mathbb{N}\), the set \(\{F_n(x) : x \in X\}\) is bounded in \(Y\)

C

the sequence \((\|F_n\|)\) is bounded in \(\mathbb{R}\)

D

the sequence \((F_n)\) is uniformly bounded on \(X\)

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6

2008 · Mathematics · Functional Analysis · Normed Spaces, Banach Spaces and Operators

Mathematics (MA) 2008

The integral equation \(x(t) = \sin t + \lambda \int_{0}^{1} (s^2 t^3 + e^{s+t} x(s)) ds, 0 \leq t \leq 1, \lambda \in \mathbb{R}, \lambda \neq 0\) has a solution for

A

all non-zero values of \(\lambda\)

B

no value of \(\lambda\)

C

only countably many positive values of \(\lambda\)

D

only countably many negative values of \(\lambda\)

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7

2008 · Mathematics · Functional Analysis · Normed Spaces, Banach Spaces and Operators

Mathematics (MA) 2008

Let \(S = \{x \in X : \|x\|_{\infty} \leq 1\}\). Then

A

\(S\) is convex and compact

B

\(S\) is not convex but compact

C

\(S\) is convex but not compact

D

\(S\) is neither convex nor compact

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8

2009 · Mathematics · Functional Analysis · Normed Spaces, Banach Spaces and Operators

Mathematics (MA) 2009

Consider the metrics d₁(f,g) = (∫_a^b |f(t) – g(t)|² dt)^1/2 and d_∞(f,g) = sup_t∈[a,b] |f(t) – g(t)| on the space X = C[a,b] of all real valued continuous functions on [a,b]. Then which of the following is TRUE?

A

Both (X, d₁) and (X, d_∞) are complete.

B

(X, d₁) is complete but (X, d_∞) is NOT complete.

C

(X, d_∞) is complete but (X, d₁) is NOT complete.

D

Both (X, d₁) and (X, d_∞) are NOT complete.

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9

2009 · Mathematics · Functional Analysis · Normed Spaces, Banach Spaces and Operators

Mathematics (MA) 2009

Let \( f : (c_{00}, \|\cdot\|_1) \to \mathbb{C} \) be a non-zero continuous linear functional. The number of Hahn-Banach extensions of \( f \) to \( (\ell^1, \|\cdot\|_1) \) is

A

one

B

two

C

three

D

infinite

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10

2009 · Mathematics · Functional Analysis · Normed Spaces, Banach Spaces and Operators

Mathematics (MA) 2009

If \( I : (\ell^1, \|\cdot\|_2) \to (\ell^1, \|\cdot\|_1) \) is the identity map, then

A

both \(I \) and \(I^{-1} \) are continuous

B

\(I \) is continuous but \(I^{-1} \) is NOT continuous

C

\(I^{-1} \) is continuous but \(I \) is NOT continuous

D

neither \(I \) nor \(I^{-1} \) is continuous

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11

2009 · Mathematics · Functional Analysis · Normed Spaces, Banach Spaces and Operators

Mathematics (MA) 2009

Let \( X \) and \( Y \) be Banach spaces and let \( T : X \to Y \) be a linear map. Consider the statements: \( P \): If \( x_n \to x \) in \( X \) then \( Tx_n \to Tx \) in \( Y \). \( Q \): If \( x_n \to x \) in \( X \) and \( Tx_n \to y \) in \( Y \) then \( Tx = y \). Then

A

\(P \) implies \(Q \) and \(Q \) implies \(P \)

B

\(P \) implies \(Q \) but \(Q \) does not imply \(P \)

C

\(Q \) implies \(P \) but \(P \) does not imply \(Q \)

D

neither \(P \) implies \(Q \) nor \(Q \) implies \(P \)

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12

2010 · Mathematics · Functional Analysis · Normed Spaces, Banach Spaces and Operators

Mathematics (MA) 2010

Let \( X \) and \( Y \) be normed linear spaces and \( \{T_n\} \) be a sequence of bounded linear operators from \( X \) to \( Y \). Consider the statements:
\( P : \{ \|T_n x\| : n \in \mathbb{N} \} \) is bounded for each \( x \in X \)
\( Q : \{ \|T_n\| : n \in \mathbb{N} \} \) is bounded
Which one of the following is correct?

A

If \(P \) implies \(Q \), then both \(X \) and \(Y \) are Banach spaces

B

If \(P \) implies \(Q \), then only one of \(X \) and \(Y \) is a Banach space

C

If \(X \) is a Banach space, then \(P \) implies \(Q \)

D

If \(Y \) is a Banach space, then \(P \) implies \(Q \)

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13

2010 · Mathematics · Functional Analysis · Normed Spaces, Banach Spaces and Operators

Mathematics (MA) 2010

Let \( X = C[0, 1] \) with the norm \( \|x\| = \int_{0}^{1} |x(t)| dt \), \( x \in C[0, 1] \) and \( \Omega = \{ f \in X^* : \|f\| = 1 \} \), where \( X^* \) denotes the dual space of \( X \). Let \( C(\Omega) \) be the linear space of continuous functions on \( \Omega \) with the norm \( \|u\| = \sup_{f \in \Omega} |u(f)| \), \( u \in C(\Omega) \). Then

A

\(X \) is linearly isometric with \(C(\Omega) \)

B

\(X \) is linearly isometric with a proper subspace of \(C(\Omega) \)

C

there does not exist a linear isometry from \(X \) into \(C(\Omega) \)

D

every linear isometry from \(X \) to \(C(\Omega) \) is onto

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14

2011 · Mathematics · Functional Analysis · Normed Spaces, Banach Spaces and Operators

Mathematics (MA) 2011

Let \( x = (x_1, x_2, \ldots) \in l^4, x \neq 0. \) For which one of the following values of \( p, \) the series \( \sum_{i=1}^{\infty} x_i y_i \) converges for every \( y = (y_1, y_2, \ldots) \in l^p \)?

A

1

B

2

C

3

D

4

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15

2011 · Mathematics · Functional Analysis · Normed Spaces, Banach Spaces and Operators

Mathematics (MA) 2011

Let \( X \) be the real normed linear space of all real sequences with finitely many non-zero terms, with supremum norm and \( T : X \to X \) be a one to one and onto linear operator defined by \( T(x_1, x_2, x_3, \ldots) = \left( x_1, \frac{x_2}{2}, \frac{x_3}{3}, \ldots \right) \). Then, which of the following is TRUE?

A

\(T \) is bounded but \(T^{-1} \) is not bounded

B

\(T \) is not bounded but \(T^{-1} \) is bounded

C

Both \(T \) and \(T^{-1} \) are bounded

D

Neither \(T \) nor \(T^{-1} \) is bounded

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16

2012 · Mathematics · Functional Analysis · Normed Spaces, Banach Spaces and Operators

Mathematics (MA) 2012

Consider the statements
P: If \(X\) is a normed linear space and \(M \subseteq X\) is a subspace, then the closure \(\overline{M}\) is also a subspace of \(X\).
Q: If \(X\) is a Banach space and \(\sum x_n\) is an absolutely convergent series in \(X\), then \(\sum x_n\) is convergent.
R: Let \(M_1\) and \(M_2\) be subspaces of an inner product space such that \(M_1 \cap M_2 = \{0\}\). Then \(\forall m_1 \in M_1, m_2 \in M_2 ; \|m_1 + m_2\|^2 = \|m_1\|^2 + \|m_2\|^2\).
S: Let \(f : X \to Y\) be a linear transformation from the Banach Space \(X\) into the Banach space \(Y\). If \(f\) is continuous, then the graph of \(f\) is always compact.
The correct statements amongst the above are:

A

P and R only

B

Q and R only

C

P and Q only

D

R and S only

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17

2013 · Mathematics · Functional Analysis · Normed Spaces, Banach Spaces and Operators

Mathematics (MA) 2013

Which of the following is FALSE?

A

\(f\) is a continuous linear functional

B

\(\|f\| \le \frac{\pi}{\sqrt{6}}\)

C

There does not exist any \(y \in c_{00}\) such that \(f(x) = \langle x, y \rangle\) for all \(x \in c_{00}\)

D

\(N^{\perp} \neq \{0\}\)

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18

2014 · Mathematics · Functional Analysis · Normed Spaces, Banach Spaces and Operators

Mathematics (MA) 2014

Which of the following statements about the spaces ℓᵖ and Lᵖ[0,1] is TRUE?

A

ℓ² ⊂ ℓ³ and L⁶[0,1] ⊂ L⁹[0,1]

B

ℓ³ ⊂ ℓ² and L⁹[0,1] ⊂ L⁶[0,1]

C

ℓ⁷ ⊂ ℓ³ and L⁶[0,1] ⊂ L⁹[0,1]

D

ℓ⁷ ⊂ ℓ³ and L⁹[0,1] ⊂ L⁶[0,1]

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19

2014 · Mathematics · Functional Analysis · Normed Spaces, Banach Spaces and Operators

Mathematics (MA) 2014

Consider C[-1,1] equipped with the supremum norm given by
||f||_∞ = sup{|f(t)| : t ∈ [-1,1]} for f ∈ C[-1,1]. Define a linear functional T on C[-1,1] by
T(f) = ∫₋₁⁰ f(t)dt - ∫₀¹ f(t)dt for all f ∈ C[-1,1]. Then the value of ||T|| is __________

Check answer

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20

2014 · Mathematics · Functional Analysis · Normed Spaces, Banach Spaces and Operators

Mathematics (MA) 2014

Let \(X = C^1[0,1]\). For each \(f \in X\), define
\(p_1(f) := \sup \{|f(t)| : t \in [0,1]\}\)
\(p_2(f) := \sup \{|f'(t)| : t \in [0,1]\}\)
\(p_3(f) := p_1(f) + p_2(f)\).
Which of the following statements is TRUE?

A

\((X, p_1)\) is a Banach space

B

\((X, p_2)\) is a Banach space

C

\((X, p_3)\) is NOT a Banach space

D

\((X, p_3)\) does NOT have denumerable basis

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