Kreyszig 2.3, Further Properties of Normed Spaces

Problem 1. Show that clc \subset l^{\infty} is a vector subspace of ll^{\infty} and so is C0C_0, the space of all sequences of scalars converging to zero.

Solution:

The space ll^{\infty} is defined as the set of all bounded sequences of real (or complex) numbers. A sequence (an)(a_n) is in ll^{\infty} if there exists a real number MM such that for every term ana_n in the sequence, anM|a_n| \leq M.

The space cc denotes the set of all convergent sequences. A sequence (an)(a_n) is in cc if it converges to some limit LL in the real (or complex) numbers.

The space C0C_0, or c0c_0 as it is often denoted, is the set of all sequences that converge to zero.

To show that cc and C0C_0 are subspaces of ll^{\infty}, we must verify the following properties for each:

  1. Non-emptiness: The subspace must contain the zero vector.

  2. Closed under vector addition: If two vectors xx and yy are in the subspace, then their sum x+yx + y must also be in the subspace.

  3. Closed under scalar multiplication: If a vector xx is in the subspace and α\alpha is any scalar, then the product αx\alpha x must also be in the subspace.

For cc (all convergent sequences):

  1. Non-emptiness: The zero sequence is in cc since it converges to zero, and it is clearly bounded.

  2. Closed under vector addition: If x,ycx, y \in c, both converge to some limits LxL_x and LyL_y, and their sum x+yx + y converges to Lx+LyL_x + L_y. Also, the sum of two bounded sequences is bounded.

  3. Closed under scalar multiplication: For any xcx \in c and scalar α\alpha, the sequence αx\alpha x converges to αLx\alpha L_x and is bounded if xx is bounded.

For C0C_0 (sequences converging to zero):

  1. Non-emptiness: C0C_0 contains the zero sequence.

  2. Closed under vector addition: The sum of two sequences in C0C_0 also converges to zero.

  3. Closed under scalar multiplication: A scalar multiple of a sequence in C0C_0 also converges to zero and is bounded.

Since both cc and C0C_0 satisfy these properties, they are both subspaces of ll^{\infty}.


Problem 2. Show that c0c_0 in Problem 1 is a closed subspace of ll^{\infty}, so that c0c_0 is complete by: (a) Theorem (Complete subspace): A subspace MM of a complete metric space XX is itself complete if and only if the set MM is closed in XX. (b) Completeness of ll^{\infty}: The space ll^{\infty}, is complete.

Solution:

To show that c0c_0 from Problem 1 is a closed subspace of ll^{\infty}, we will use the theorem provided and the fact that ll^{\infty} is complete.

Step 1: Use the Theorem (Complete Subspace)

The theorem states that a subspace MM of a complete metric space XX is complete if and only if MM is closed in XX. Therefore, we must demonstrate that c0c_0 is closed in ll^{\infty}.

Step 2: Show that c0c_0 is closed in ll^{\infty}

A subset of a metric space is closed if it contains all of its limit points. To prove that c0c_0 is closed, we need to show that if a sequence of elements in c0c_0 converges to some limit within ll^{\infty}, then this limit is also in c0c_0.

Suppose (xn)(x_n) is a sequence of sequences in c0c_0 that converges to some sequence xx in ll^{\infty}. We need to show that xx is also in c0c_0. This means that xx must converge to zero.

Since (xn)(x_n) converges to xx in ll^{\infty}, for every ϵ>0\epsilon > 0, there exists an NN such that for all nNn \geq N, the sequences xnx_n are within ϵ\epsilon of xx in the supremum norm, i.e.,

supkN(xn)kxk<ϵ. \sup_{k \in \mathbb{N}} |(x_n)_k - x_k| < \epsilon.

Each xnx_n is in c0c_0, meaning that for each xnx_n and for every ϵ>0\epsilon > 0, there exists an MM (which can depend on nn) such that for all kMk \geq M, (xn)k<ϵ|(x_n)_k| < \epsilon.

As nn \rightarrow \infty, xnx_n converges to xx and since each xnx_n gets arbitrarily close to zero for large enough indices, xx must also get arbitrarily close to zero for large enough indices. This means that xx converges to zero and thus xc0x \in c_0.

Step 3: Apply the Completeness of ll^{\infty}

Since ll^{\infty} is a complete metric space and c0c_0 is closed in ll^{\infty}, by the theorem, c0c_0 is also complete.

By showing that c0c_0 is a closed subset of the complete space ll^{\infty}, we have shown that c0c_0 is a complete subspace of ll^{\infty}.


Problem 3. Problem Statement: In ll^{\infty}, let YY be the subset of all sequences with only finitely many nonzero terms. Show that YY is a subspace of ll^{\infty} but not a closed subspace.

Solution:

To demonstrate that YY is a subspace of ll^{\infty}, we must verify that YY satisfies the three properties of a vector subspace:

  1. Non-emptiness: YY contains the zero vector.

  2. Closed under vector addition: If two vectors xx and yy are in YY, then their sum x+yx + y must also be in YY.

  3. Closed under scalar multiplication: If a vector xx is in YY and α\alpha is any scalar, then the product αx\alpha x must also be in YY.

Let's examine each property:

  1. Non-emptiness: The zero sequence, where every term is zero, is a sequence with finitely many nonzero terms (specifically, none), so YY contains the zero vector.

  2. Closed under vector addition: If xx and yy are in YY, they each have only finitely many nonzero terms. The sum x+yx + y will also have only finitely many nonzero terms because the nonzero terms can only occur at the indices where xx or yy (or both) have nonzero terms. Therefore, x+yx + y is also in YY.

  3. Closed under scalar multiplication: If xx is in YY and α\alpha is any scalar, multiplying xx by α\alpha will not introduce any new nonzero terms beyond those already present in xx. Therefore, αx\alpha x will also have only finitely many nonzero terms and is in YY.

Since YY satisfies all three properties, it is a subspace of ll^{\infty}.

To show that YY is not a closed subspace, we need to find a sequence of elements in YY that converges to a limit not in YY. This limit will be a sequence with infinitely many nonzero terms, demonstrating that YY does not contain all its limit points, and hence it is not closed.

Consider the sequence of sequences (y(n))(y^{(n)}) defined by:

y(n)=(1,12,13,,1n,0,0,0,) y^{(n)} = (1, \frac{1}{2}, \frac{1}{3}, \ldots, \frac{1}{n}, 0, 0, 0, \ldots)

Each y(n)y^{(n)} is in YY because it has only nn nonzero terms. Now, consider the sequence yy defined by:

y=(1,12,13,) y = (1, \frac{1}{2}, \frac{1}{3}, \ldots)

The sequence yy is not in YY because it has infinitely many nonzero terms. However, (y(n))(y^{(n)}) converges to yy in the ll^{\infty} norm because for every ϵ>0\epsilon > 0, there exists an NN such that for all nNn \geq N, the tail of the sequence yy (from nn onward) is bounded above by ϵ\epsilon.

Therefore, the limit of the convergent sequence (y(n))(y^{(n)}) is not in YY, showing that YY is not closed in ll^{\infty}.


Problem 4. In a normed space XX, show that vector addition and multiplication by scalars are continuous operations with respect to the norm; that is, the mappings defined by (x,y)x+y(x, y) \mapsto x+y and (α,x)αx(\alpha, x) \mapsto \alpha x are continuous.

Solution:

Continuity of Vector Addition

Let (xn)(x_n) and (yn)(y_n) be sequences in XX such that xnxx_n \to x and ynyy_n \to y as nn \to \infty. We need to show that xn+ynx+yx_n + y_n \to x + y. By the definition of convergence in a normed space, for every ϵ>0\epsilon > 0, there exist N1,N2NN_1, N_2 \in \mathbb{N} such that for all nN1n \geq N_1, xnx<ϵ2\|x_n - x\| < \frac{\epsilon}{2} and for all nN2n \geq N_2, yny<ϵ2\|y_n - y\| < \frac{\epsilon}{2}.

Let N=max{N1,N2}N = \max\{N_1, N_2\}. Then for all nNn \geq N, we have:

(xn+yn)(x+y)=(xnx)+(yny)xnx+yny<ϵ2+ϵ2=ϵ \| (x_n + y_n) - (x + y) \| = \| (x_n - x) + (y_n - y) \| \leq \|x_n - x\| + \|y_n - y\| < \frac{\epsilon}{2} + \frac{\epsilon}{2} = \epsilon

The inequality follows from the triangle inequality of the norm. Since ϵ\epsilon was arbitrary, this shows that xn+ynx+yx_n + y_n \to x + y, and thus vector addition is continuous.

Continuity of Scalar Multiplication

Let (αn)(\alpha_n) be a sequence of scalars converging to α\alpha, and let (xn)(x_n) be a sequence in XX such that xnxx_n \to x. We need to show that αnxnαx\alpha_n x_n \to \alpha x. For every ϵ>0\epsilon > 0, there exist N1,N2NN_1, N_2 \in \mathbb{N} such that for all nN1n \geq N_1, αnα<ϵ2(x+1)|\alpha_n - \alpha| < \frac{\epsilon}{2(\|x\|+1)} and for all nN2n \geq N_2, xnx<ϵ2(α+1)\|x_n - x\| < \frac{\epsilon}{2(\|\alpha\|+1)}.

Let N=max{N1,N2}N = \max\{N_1, N_2\}. Then for all nNn \geq N, we have:

αnxnαx=αnxnαnx+αnxαxαn(xnx)+(αnα)x \| \alpha_n x_n - \alpha x \| = \| \alpha_n x_n - \alpha_n x + \alpha_n x - \alpha x \| \leq \| \alpha_n (x_n - x) \| + \| (\alpha_n - \alpha) x \|

Using the properties of the norm and the convergence of αn\alpha_n and xnx_n, we further obtain:

αnxnαxαnxnx+αnαx<(α+1)ϵ2(α+1)+ϵ2=ϵ \| \alpha_n x_n - \alpha x \| \leq |\alpha_n| \| x_n - x \| + |\alpha_n - \alpha| \| x \| < (\|\alpha\|+1) \frac{\epsilon}{2(\|\alpha\|+1)} + \frac{\epsilon}{2} = \epsilon

Since ϵ\epsilon was arbitrary, this shows that αnxnαx\alpha_n x_n \to \alpha x, and thus scalar multiplication is continuous.

Hence, in a normed space XX, both vector addition and scalar multiplication are continuous with respect to the norm.


Problem 5. Show that xnxx_n \to x and ynyy_n \to y implies xn+ynx+yx_n + y_n \to x + y. Show that αnα\alpha_n \to \alpha and xnxx_n \to x implies αnxnαx\alpha_n x_n \to \alpha x.

Solution:

Continuity of Vector Addition

Given xnxx_n \to x and ynyy_n \to y, we need to demonstrate that xn+ynx+yx_n + y_n \to x + y.

By the definition of convergence, for every ϵ>0\epsilon > 0, there exists an N1N_1 such that for all nN1n \geq N_1, xnx<ϵ2\|x_n - x\| < \frac{\epsilon}{2}. Similarly, there exists an N2N_2 such that for all nN2n \geq N_2, yny<ϵ2\|y_n - y\| < \frac{\epsilon}{2}.

Let N=max(N1,N2)N = \max(N_1, N_2). Then for all nNn \geq N:

(xn+yn)(x+y)=(xnx)+(yny)xnx+yny<ϵ2+ϵ2=ϵ. \| (x_n + y_n) - (x + y) \| = \| (x_n - x) + (y_n - y) \| \leq \|x_n - x\| + \|y_n - y\| < \frac{\epsilon}{2} + \frac{\epsilon}{2} = \epsilon.

This proves that xn+ynx+yx_n + y_n \to x + y, confirming the continuity of vector addition.

Continuity of Scalar Multiplication

Given αnα\alpha_n \to \alpha and xnxx_n \to x, we need to show that αnxnαx\alpha_n x_n \to \alpha x.

For every ϵ>0\epsilon > 0, there exists an N1N_1 such that for all nN1n \geq N_1, αnα<ϵ2(x+1)|\alpha_n - \alpha| < \frac{\epsilon}{2(\|x\| + 1)} (assuming x0x \neq 0, otherwise the result is trivial). Also, there exists an N2N_2 such that for all nN2n \geq N_2, xnx<ϵ2(α+1)\|x_n - x\| < \frac{\epsilon}{2(|\alpha| + 1)}.

Let N=max(N1,N2)N = \max(N_1, N_2). Then for all nNn \geq N:

αnxnαx=αnxnαnx+αnxαxαnxnx+αnαx. \| \alpha_n x_n - \alpha x \| = \| \alpha_n x_n - \alpha_n x + \alpha_n x - \alpha x \| \leq |\alpha_n| \| x_n - x \| + |\alpha_n - \alpha| \| x \|.

Using the convergence criteria and the norm properties, we get:

αnxnx<(α+1)ϵ2(α+1)=ϵ2, |\alpha_n| \| x_n - x \| < (|\alpha| + 1) \frac{\epsilon}{2(|\alpha| + 1)} = \frac{\epsilon}{2},

and

αnαx<ϵ2(x+1)xϵ2. |\alpha_n - \alpha| \| x \| < \frac{\epsilon}{2(\|x\| + 1)} \|x\| \leq \frac{\epsilon}{2}.

Summing these inequalities gives:

αnxnαx<ϵ. \| \alpha_n x_n - \alpha x \| < \epsilon.

This confirms that αnxnαx\alpha_n x_n \to \alpha x, establishing the continuity of scalar multiplication.


Problem 6. Show that the closure Yˉ\bar{Y} of a subspace YY of a normed space XX is again a vector subspace.

Solution:

To show that the closure Y\overline{Y} of a subspace YY is a vector subspace, we need to verify that it satisfies the properties of a vector subspace:

Non-emptiness: The closure Y\overline{Y} must contain the zero vector. Since YY is a subspace, it contains the zero vector 00. The closure of a set contains all the limit points of that set, and since 00 is in YY and is its own limit, 00 is also in Y\overline{Y}.

Closed under vector addition: If xx and yy are in Y\overline{Y}, then x+yx + y must also be in Y\overline{Y}. Let xx and yy be in Y\overline{Y}. By the definition of closure, for every ϵ>0\epsilon > 0, there exist points xYx' \in Y and yYy' \in Y such that xx<ϵ2\|x - x'\| < \frac{\epsilon}{2} and yy<ϵ2\|y - y'\| < \frac{\epsilon}{2}. Since YY is a subspace and therefore closed under addition, x+yx' + y' is in YY.

Consider x+yx + y and x+yx' + y'. We have:

(x+y)(x+y)=(xx)+(yy)xx+yy<ϵ2+ϵ2=ϵ. \| (x + y) - (x' + y') \| = \| (x - x') + (y - y') \| \leq \|x - x'\| + \|y - y'\| < \frac{\epsilon}{2} + \frac{\epsilon}{2} = \epsilon.

This inequality shows that for every point x+yx + y in Y\overline{Y}, we can find a point x+yx' + y' in YY such that x+yx + y is as close as we wish to x+yx' + y', which means x+yx + y is a limit point of YY and hence in Y\overline{Y}.

Closed under scalar multiplication: If xx is in Y\overline{Y} and α\alpha is a scalar, then αx\alpha x must also be in Y\overline{Y}. Let xx be in Y\overline{Y} and let α\alpha be any scalar. By the definition of closure, for every ϵ>0\epsilon > 0, there exists a point xYx' \in Y such that xx<ϵα\|x - x'\| < \frac{\epsilon}{|\alpha|} if α0\alpha \neq 0 (if α=0\alpha = 0, the result is trivial since 0x=00 \cdot x = 0 is in YY and hence in Y\overline{Y}).

Since YY is a subspace, it is closed under scalar multiplication, so αx\alpha x' is in YY. Consider αx\alpha x and αx\alpha x'. We have:

αxαx=αxx<αϵα=ϵ. \| \alpha x - \alpha x' \| = |\alpha| \| x - x' \| < |\alpha| \cdot \frac{\epsilon}{|\alpha|} = \epsilon.

This inequality shows that for every point αx\alpha x in Y\overline{Y}, we can find a point αx\alpha x' in YY such that αx\alpha x is as close as we wish to αx\alpha x', which means αx\alpha x is a limit point of YY and hence in Y\overline{Y}.

Therefore, the closure Y\overline{Y} of a subspace YY of a normed space XX satisfies all the properties of a vector subspace and is thus itself a vector subspace of XX.


Problem 7. Show that convergence of y1+y2+y3+\|\mathbf{y}_1\| + \|\mathbf{y}_2\| + \|\mathbf{y}_3\| + \ldots may not imply convergence of y1+y2+y3+\mathbf{y}_1 + \mathbf{y}_2 + \mathbf{y}_3 + \ldots. Hint: Consider y\mathbf{y} in Prob. 3 and (yn)(\mathbf{y}_n), where yn=(ηj(n))\mathbf{y}_n = (\eta_j^{(n)}), ηn(n)=1/n2\eta_n^{(n)} = 1/n^2, ηj(n)=0\eta_j^{(n)} = 0 for all jnj \neq n.

Solution:

To demonstrate the statement, we'll consider a sequence in the space ll^\infty of all bounded sequences of scalars, which is the space mentioned in Problem 3.

We'll construct a specific example using the hint provided, which involves sequences with only one non-zero term whose magnitude is 1n2\frac{1}{n^2}. This example will show that the series of norms converges (absolute convergence), but the series of vectors does not converge in the ll^\infty space.

Construction:

Let yny_n be a sequence in ll^\infty defined by yn=(ηj(n))y_n = (\eta_j^{(n)}) where:

ηj(n)={1n2if j=n0if jn \eta_j^{(n)} = \begin{cases} \frac{1}{n^2} & \text{if } j = n \\ 0 & \text{if } j \neq n \end{cases}

This sequence yny_n has only the nn-th term non-zero and equal to 1n2\frac{1}{n^2}, and all other terms are zero.

Absolute convergence of norms:

Consider the series of norms n=1yn\sum_{n=1}^\infty \|y_n\|. Since yn=1n2\|y_n\| = \frac{1}{n^2} for each nn, the series is:

n=1yn=n=11n2 \sum_{n=1}^\infty \|y_n\| = \sum_{n=1}^\infty \frac{1}{n^2}

The series n=11n2\sum_{n=1}^\infty \frac{1}{n^2} is known to converge (it's a p-series with p=2p = 2, which converges for p>1p > 1).

Lack of convergence of the vector series:

Now consider the series of vectors n=1yn\sum_{n=1}^\infty y_n. The nn-th partial sum of this series is:

Sn=k=1nyk=(1,14,19,,1n2,0,0,) S_n = \sum_{k=1}^n y_k = (1, \frac{1}{4}, \frac{1}{9}, \ldots, \frac{1}{n^2}, 0, 0, \ldots)

Each partial sum SnS_n is a sequence in ll^\infty where the first nn terms are the reciprocals of the squares of the natural numbers, and the rest are zeros.

The limit of the partial sums SnS_n as nn \to \infty, if it exists, would be the sequence:

S=(1,14,19,,1n2,) S = (1, \frac{1}{4}, \frac{1}{9}, \ldots, \frac{1}{n^2}, \ldots)

The sequence SS represents the harmonic series of squares, which does not converge in the ll^\infty space, because it's not a bounded sequence. Each term in the sequence SS is a positive number, and there are infinitely many terms, so the sequence does not converge to a point in ll^\infty (which requires boundedness).

Conclusion:

We have shown that while the series of norms n=1yn\sum_{n=1}^\infty \|y_n\| converges, the series of vectors n=1yn\sum_{n=1}^\infty y_n does not converge in the ll^\infty space. This example illustrates that absolute convergence of the norms does not imply convergence of the series of vectors in the ll^\infty space.


Problem 8. Problem Statement: In a normed space XX, if absolute convergence of any series always implies convergence of that series, show that XX is complete.

Proof:

  1. Absolute Convergence Implies Convergence: By hypothesis, if a series n=1xn\sum_{n=1}^\infty x_n in XX is absolutely convergent, meaning that n=1xn\sum_{n=1}^\infty \|x_n\| converges, then the series n=1xn\sum_{n=1}^\infty x_n itself converges in XX.

  2. Cauchy Criterion for Series: A series n=1xn\sum_{n=1}^\infty x_n converges if and only if the sequence of partial sums Sm=n=1mxnS_m = \sum_{n=1}^m x_n is a Cauchy sequence.

  3. Absolute Convergence and Cauchy Sequences: Suppose n=1xn\sum_{n=1}^\infty x_n is absolutely convergent. Then for every ε>0\varepsilon > 0, there exists NNN \in \mathbb{N} such that for all m>nNm > n \geq N, we have k=nmxk<ε\sum_{k=n}^m \|x_k\| < \varepsilon because the series of norms is convergent and hence satisfies the Cauchy criterion.

  4. Implication for Partial Sums: The property above implies that the sequence of partial sums (Sm)(S_m) is Cauchy. To see this, note that for m>nNm > n \geq N,

    SmSn=k=n+1mxkk=n+1mxk<ε. \|S_m - S_n\| = \left\|\sum_{k=n+1}^m x_k\right\| \leq \sum_{k=n+1}^m \|x_k\| < \varepsilon.

    This inequality holds because the norm is subadditive (it satisfies the triangle inequality).

  5. Completeness of X: If (Sm)(S_m) is a Cauchy sequence in XX and XX is a space where absolute convergence implies convergence, then (Sm)(S_m) must converge in XX because it is absolutely convergent.

  6. Conclusion: Since every Cauchy sequence in XX converges in XX, XX is complete. Hence, XX is a Banach space.

The key point here is the equivalence of the Cauchy criterion for series convergence and the completeness of the space. The hypothesis that absolute convergence implies convergence ensures that Cauchy sequences of partial sums always converge, which is precisely the definition of a complete space. Solution:

The key point here is the equivalence of the Cauchy criterion for series convergence and the completeness of the space. The hypothesis that absolute convergence implies convergence ensures that Cauchy sequences of partial sums always converge, which is precisely the definition of a complete space.

Completeness of XX:

To say that XX is complete means that every Cauchy sequence in XX converges to a limit within XX. Now, let's consider any Cauchy sequence (xn)(x_n) in XX. By the property of normed spaces, we can form a series n=1(xn+1xn)\sum_{n=1}^\infty (x_{n+1} - x_n). This series is absolutely convergent if the series of norms n=1xn+1xn\sum_{n=1}^\infty \|x_{n+1} - x_n\| converges.

Since (xn)(x_n) is a Cauchy sequence, for every ε>0\varepsilon > 0, there exists an NN such that for all m>nNm > n \geq N, the distance between xnx_n and xmx_m is less than ε\varepsilon. Formally, xmxn<ε\|x_m - x_n\| < \varepsilon.

The property of being a Cauchy sequence suggests that the series of differences (xn+1xn)(x_{n+1} - x_n) has terms that become arbitrarily small as nn increases. In other words, the series n=Nxn+1xn\sum_{n=N}^\infty \|x_{n+1} - x_n\| has terms that decrease and approach zero, implying that the series of norms is convergent.

Given that absolute convergence of a series in XX implies its convergence, we can conclude that the series n=1(xn+1xn)\sum_{n=1}^\infty (x_{n+1} - x_n) converges in XX. The convergence of this series means that the sequence of partial sums, which corresponds to the sequence (xn)(x_n) up to an initial segment, converges to a limit in XX.

Therefore, the original Cauchy sequence (xn)(x_n) must also converge in XX, because its behavior at infinity is captured by the series formed by its successive differences. Since every Cauchy sequence in XX has a limit in XX, we conclude that XX is complete.

In summary, the condition that absolute convergence implies convergence in XX allows us to transform the Cauchy criterion for sequences into a condition on series. Since this condition guarantees convergence for all absolutely convergent series—and hence for all Cauchy sequences—it follows that XX is a complete normed space, or a Banach space.

Solution:

This proof leverages the fundamental property of normed spaces: a space is complete if every Cauchy sequence converges within the space. The given condition, that absolute convergence implies convergence, is used to show that Cauchy sequences, constructed from series of vectors in the space, converge. This implies that the space must be complete, as all such Cauchy sequences have a limit in the space, satisfying the definition of a Banach space.


Problem 9. Show that in a Banach space, an absolutely convergent series is convergent.

Detailed Proof:

Let (X,)(X, \|\cdot\|) be a Banach space. Suppose we have a series n=1xn\sum_{n=1}^\infty x_n in XX that is absolutely convergent. By definition, this means that the series n=1xn\sum_{n=1}^\infty \|x_n\| converges in the real numbers.

  1. Definition of Absolute Convergence: The series n=1xn\sum_{n=1}^\infty \|x_n\| is said to be absolutely convergent if the sum of the norms, which are real numbers, is a convergent series in R\mathbb{R}, i.e., there exists a real number LL such that for every ϵ>0\epsilon > 0, there is an integer NN such that for all nNn \geq N, it holds that

    k=N+1nxkL<ϵ. \left|\sum_{k=N+1}^n \|x_k\| - L\right| < \epsilon.
  2. Partial Sums as a Sequence: Define the nn-th partial sum SnS_n of the series n=1xn\sum_{n=1}^\infty x_n by Sn=k=1nxkS_n = \sum_{k=1}^n x_k. The sequence (Sn)(S_n) is a sequence of elements in XX.

  3. Partial Sums Are Cauchy: To show that (Sn)(S_n) is a Cauchy sequence, consider any ϵ>0\epsilon > 0. Since the series of norms converges, there exists an integer NN such that for all m,nNm, n \geq N with m<nm < n, we have

    k=m+1nxk<ϵ. \sum_{k=m+1}^n \|x_k\| < \epsilon.

    Now, consider the difference between the nn-th and mm-th partial sums:

    SnSm=k=m+1nxkk=m+1nxk, \|S_n - S_m\| = \left\|\sum_{k=m+1}^n x_k\right\| \leq \sum_{k=m+1}^n \|x_k\|,

    where we used the triangle inequality for norms. Given our choice of NN, for m,nNm, n \geq N, this implies

    SnSm<ϵ. \|S_n - S_m\| < \epsilon.

    This is the Cauchy criterion for sequences in a normed space: for any ϵ>0\epsilon > 0, there exists an NN such that for all m,nNm, n \geq N, the norm of the difference between the nn-th and mm-th terms of the sequence is less than ϵ\epsilon.

  4. Convergence of Cauchy Sequences in Banach Spaces: A Banach space is, by definition, a complete normed vector space. Completeness means that every Cauchy sequence in the space converges to a limit within the space. Since we have established that (Sn)(S_n) is a Cauchy sequence, it must converge to some limit SS in XX.

  5. Conclusion: The limit SS to which the sequence (Sn)(S_n) converges is the sum of the series n=1xn\sum_{n=1}^\infty x_n. Therefore, the series converges in XX, and we have demonstrated that an absolutely convergent series in a Banach space is indeed convergent.

Solution:

This detailed proof walks through the concepts of absolute convergence, the properties of Cauchy sequences, and the completeness of Banach spaces to conclusively show that an absolutely convergent series in a Banach space must converge. This result is a cornerstone of functional analysis and underscores the robustness of Banach spaces for analytical purposes.